1. The document is about trigonometric limits and contains 7 pages with 34 solved limits. It uses fundamental limit laws and some techniques like l'Hôpital's rule to solve the limits. Some of the limits involve expressions with sinx/x, tgx/x, secx, etc as x approaches 0 or other values.
2. The solutions show applying limit laws to transform indeterminate forms like 0/0 into determinate limits equal to constants like 1, 2/3, etc. Some limits are solved by rewriting the expressions in terms of other trig functions whose limits are known.
3. Most of the limits have solutions that evaluate to simple constants, rational numbers or trig
This document contains a 7-page document with 34 solved trigonometric limit problems. The problems use fundamental limit laws and techniques such as factoring, simplifying, and applying standard trigonometric limits to find the value of various trigonometric function limits as the variable approaches specific values.
This document provides solutions to calculating the derivative functions of various given functions. It includes:
1) Finding the derivative functions of polynomials, rational functions, exponential functions, logarithmic functions, trigonometric functions, and composite functions.
2) The solutions provide the step-by-step work and final derivative function for each problem.
3) There are over 25 problems covered across multiple pages with the aim of teaching calculation of derivative functions.
This document discusses analyzing quadratic functions of the form y = a(x - p)2 + q. It provides examples of determining the vertex, domain and range, direction of opening, axis of symmetry, and x- and y-intercepts of quadratic functions.
1. The limit as x approaches 4 of x4-16 is 0. When factored, the expression becomes (x-4)(x+4)(x2+4) which equals 0 as x approaches 4.
2. The limit as x approaches infinity of x7-x2+1 is 1. When factored, the leading terms are x7 for both the top and bottom expressions, which equals 1 as x approaches infinity.
3. The limit as x approaches -1 of x2-1 is 0. When factored, the expression becomes (x+1)(x-1) which equals 0 as the factors are 0 when x is -1.
The document discusses Taylor series expansions (desarrollos limitados) and provides 15 examples of applying the Taylor formula to common functions like exponential, logarithmic, trigonometric and hyperbolic functions. Specifically, it gives the Taylor series expansion of each function centered around 0 and in terms of powers of x, along with the little-oh notation describing the remainder term.
Implicit differentiation allows us to find slopes of lines tangent to curves that are not graphs of functions. Almost all of the time (yes, that is a mathematical term!) we can assume the curve comprises the graph of a function and differentiate using the chain rule.
This document contains a 7-page document with 34 solved trigonometric limit problems. The problems use fundamental limit laws and techniques such as factoring, simplifying, and applying standard trigonometric limits to find the value of various trigonometric function limits as the variable approaches specific values.
This document provides solutions to calculating the derivative functions of various given functions. It includes:
1) Finding the derivative functions of polynomials, rational functions, exponential functions, logarithmic functions, trigonometric functions, and composite functions.
2) The solutions provide the step-by-step work and final derivative function for each problem.
3) There are over 25 problems covered across multiple pages with the aim of teaching calculation of derivative functions.
This document discusses analyzing quadratic functions of the form y = a(x - p)2 + q. It provides examples of determining the vertex, domain and range, direction of opening, axis of symmetry, and x- and y-intercepts of quadratic functions.
1. The limit as x approaches 4 of x4-16 is 0. When factored, the expression becomes (x-4)(x+4)(x2+4) which equals 0 as x approaches 4.
2. The limit as x approaches infinity of x7-x2+1 is 1. When factored, the leading terms are x7 for both the top and bottom expressions, which equals 1 as x approaches infinity.
3. The limit as x approaches -1 of x2-1 is 0. When factored, the expression becomes (x+1)(x-1) which equals 0 as the factors are 0 when x is -1.
The document discusses Taylor series expansions (desarrollos limitados) and provides 15 examples of applying the Taylor formula to common functions like exponential, logarithmic, trigonometric and hyperbolic functions. Specifically, it gives the Taylor series expansion of each function centered around 0 and in terms of powers of x, along with the little-oh notation describing the remainder term.
Implicit differentiation allows us to find slopes of lines tangent to curves that are not graphs of functions. Almost all of the time (yes, that is a mathematical term!) we can assume the curve comprises the graph of a function and differentiate using the chain rule.
This document lists important trigonometric identities and formulas. It covers fundamental trigonometric identities involving sine, cosine, tangent, cotangent and secant. It also includes formulas for adding and subtracting angles, formulas for trigonometric products, and formulas involving double angles. There are over 30 formulas presented across multiple categories of trigonometric relationships.
This document lists various trigonometric identities including:
1) Definitions of trig functions like sine, cosine, tangent, cotangent, secant, and cosecant
2) Identities involving negative angles
3) Sum and difference identities for trig functions of summed or subtracted angles
4) Double angle and half angle identities
5) Product-to-sum and sum-to-product identities
6) Pythagorean identities relating trig functions to their squares
1) The document describes a series of steps involving binary, octal, and decimal numbers. Various operations like addition, subtraction, and conversion between number bases are shown.
2) Examples include converting the decimal number 36.541 to octal, adding octal numbers, and performing octal subtraction.
3) Formulas are provided for converting between binary and decimal representations of numbers. Operations like AND, OR, and XOR are also demonstrated between binary digits.
Matematika Ekonomi Diferensiasi fungsi sederhanalia170494
This document discusses differentiation rules for simple functions including:
- Constant functions have a derivative of 0
- Polynomial functions have derivatives that are the polynomial with the exponent decreased by 1 and multiplied by the exponent
- The product rule, quotient rule, and chain rule for differentiation
- Examples of applying these rules to differentiate a variety of functions
Lesson 30: Duality In Linear Programmingguest463822
Every linear programming problem has a dual problem, which in many cases has an interesting interpretation. The original ("primal") problem and the dual problem have the same extreme value.
The Bresenham's line drawing algorithm is an incremental scan conversion algorithm that uses only integer calculations. It works by calculating the difference between the ideal y value and the actual plotted pixel y value at each x value, and determining whether to plot the pixel above or below the line based on whether this difference is positive or negative. The algorithm takes the line endpoints as input, calculates the change in x and y per step, sets an initial decision parameter, and then iteratively plots pixels, updating the decision parameter to determine whether to plot above or below the line at each step.
The document discusses properties of limits of functions in algebra. It presents 9 properties of limits, including: (1) the limit of a constant k is equal to k; (2) the limit of x as x approaches a is equal to a; (3) the limit of kf(x) is equal to k times the limit of f(x); (4) the limit of the sum of two functions is equal to the sum of their individual limits. It also provides examples of calculating limits using these properties, such as finding the limit of 7x - 4 as x approaches 2.
Lesson 8: Derivatives of Logarithmic and Exponential Functions (worksheet sol...Matthew Leingang
This document provides solutions to derivatives of exponential, logarithmic, and other functions. It includes:
1) The derivatives of functions such as y=e^2x, y=6^x, y=ln(x^3 + 9), and y=log_3(e^x).
2) Using logarithmic differentiation to find the derivatives of functions like y=x^x^2-1 and y=(x-1)(x-2)(x-3).
3) Taking the derivative of functions involving logarithms, exponents, and square roots such as y=sin^2(x)+2sin(x) and y=x(x-1)^3/
The document presents a linear programming problem to maximize return from investing Rs. 1,00,000 in two stock portfolios. The maximum investment allowed in each portfolio is Rs. 75,000. Portfolio 1 has a 10% average return and risk rating of 4, while Portfolio 2 has 20% return and risk rating of 9. The objective is to maximize return Z = 0.10X1 + 0.20X2, subject to the constraints that total investment cannot exceed Rs. 1,00,000, individual investments are at most Rs. 75,000 each, risk rating must be below 6, and average return must be at least 12%. The optimal solution is to invest Rs. 60,000 in Port
1) The document provides definitions and formulas for calculating derivatives, including the derivative of a function at a point, the derivative function, and common derivatives of basic functions like polynomials, exponentials, logarithms, trigonometric functions, and inverse trigonometric functions.
2) Examples are given of calculating derivatives using the definition of the derivative, for functions like f(x)=3x^2, f(x)=x+1, and f(x)=1/x.
3) Rules are listed for calculating the derivatives of sums, products, quotients of functions using properties of derivatives.
1) An antiderivative of a function f(x) is any function F(x) whose derivative is equal to f(x).
2) The general antiderivative of a function f(x) is written as F(x) + C, where F(x) is a particular antiderivative and C is an arbitrary constant.
3) The indefinite integral notation ∫f(x)dx represents the entire family of antiderivatives for a function f(x), since each value of C defines a different antiderivative.
This document discusses various integral formulas for simple functions including:
1) Potential integrals from a to x of the form ∫f(x)dx
2) Logarithmic integrals of the form ∫f(x)dx/x
3) Exponential integrals of the form ∫e^f(x)dx
4) Trigonometric integrals involving sine, cosine, and tangent functions.
This document summarizes the solution to an exercise with three parts:
1) Part (a) finds the probability density function f(x) of a random variable X based on its integral from -infinity to infinity being 1. It determines that f(x) = 2 and a = 2.
2) Part (b) calculates the expected value E(x) of X by integrating x*f(x) from 0 to 1. It determines the expected value is 1/3.
3) Part (c) calculates the variance V(X) of X by finding its expected value E(X2) and subtracting the square of its expected value. It determines the variance is 1/
The document describes the midpoint circle algorithm for drawing circles on a pixel screen. It explains how the algorithm determines the midpoint between the next two possible consecutive pixels and checks if the midpoint is inside or outside the circle to determine which pixel to illuminate. It provides the mathematical equations and steps used to iteratively calculate the x and y coordinates of each pixel on the circle. The algorithm is implemented in a C++ program to draw a circle on a graphics screen.
The document is a lesson on continuity and infinite limits. It defines infinite limits, including limits approaching positive or negative infinity. It provides examples of evaluating limits at points where a function is not continuous. It also outlines several rules of thumb for manipulating infinite limits, such as the sum or product of an infinite limit with a finite limit being infinite. The document cautions that limits of indeterminate forms like 0×∞ or ∞-∞ require closer examination rather than following rules of thumb. It provides an example of rationalizing an expression to put it in a form where limit laws can be applied.
The document defines:
1) Powers and exponentiation, including rules for multiplying, dividing, and raising powers of numbers.
2) Examples are provided to illustrate the rules.
3) It notes that care must be taken with negative bases and even/odd exponents.
The summary provides the high level definition of powers/exponentiation and notes some key rules and examples are given to illustrate, highlighting the need to consider signs with negative bases. It does not include details of the specific examples or problems shown in the document.
This document provides a methodology for solving definite and indefinite integrals of various types, including simple, logarithmic, exponential, trigonometric, and their inverses. It contains over 40 examples of integrals worked out step-by-step, covering the basic rules for evaluating indefinite integrals of functions like polynomials, trigonometric functions, exponentials, and their inverses.
1. The document provides examples of limits calculations and concepts.
2. Key steps in limit calculations are presented such as evaluating one-sided limits separately if they differ and using algebraic manipulation to simplify expressions before taking the limit.
3. Problem sets with solutions demonstrate various types of limits, including one-sided limits, limits at infinity, limits of rational functions, and limits of trigonometric functions.
This document contains an instructor's resource manual for limits concepts with examples and problem sets. It includes definitions of limits, worked examples of limit calculations, and limit problems to solve.
This document contains an instructor's resource manual for limits concepts with examples and problem sets. It includes definitions of limits, worked examples of limit calculations, and limit problems to solve.
This document provides the solutions manual for Trigonometry 10th Edition by Larson. It includes solutions for all exercises in Chapter 2 on Analytic Trigonometry. The chapter covers fundamental trigonometric identities, verifying identities, solving trigonometric equations, sum and difference formulas, and multiple-angle and product-to-sum formulas. The solutions provide step-by-step workings to arrive at the answers for each problem.
This document lists important trigonometric identities and formulas. It covers fundamental trigonometric identities involving sine, cosine, tangent, cotangent and secant. It also includes formulas for adding and subtracting angles, formulas for trigonometric products, and formulas involving double angles. There are over 30 formulas presented across multiple categories of trigonometric relationships.
This document lists various trigonometric identities including:
1) Definitions of trig functions like sine, cosine, tangent, cotangent, secant, and cosecant
2) Identities involving negative angles
3) Sum and difference identities for trig functions of summed or subtracted angles
4) Double angle and half angle identities
5) Product-to-sum and sum-to-product identities
6) Pythagorean identities relating trig functions to their squares
1) The document describes a series of steps involving binary, octal, and decimal numbers. Various operations like addition, subtraction, and conversion between number bases are shown.
2) Examples include converting the decimal number 36.541 to octal, adding octal numbers, and performing octal subtraction.
3) Formulas are provided for converting between binary and decimal representations of numbers. Operations like AND, OR, and XOR are also demonstrated between binary digits.
Matematika Ekonomi Diferensiasi fungsi sederhanalia170494
This document discusses differentiation rules for simple functions including:
- Constant functions have a derivative of 0
- Polynomial functions have derivatives that are the polynomial with the exponent decreased by 1 and multiplied by the exponent
- The product rule, quotient rule, and chain rule for differentiation
- Examples of applying these rules to differentiate a variety of functions
Lesson 30: Duality In Linear Programmingguest463822
Every linear programming problem has a dual problem, which in many cases has an interesting interpretation. The original ("primal") problem and the dual problem have the same extreme value.
The Bresenham's line drawing algorithm is an incremental scan conversion algorithm that uses only integer calculations. It works by calculating the difference between the ideal y value and the actual plotted pixel y value at each x value, and determining whether to plot the pixel above or below the line based on whether this difference is positive or negative. The algorithm takes the line endpoints as input, calculates the change in x and y per step, sets an initial decision parameter, and then iteratively plots pixels, updating the decision parameter to determine whether to plot above or below the line at each step.
The document discusses properties of limits of functions in algebra. It presents 9 properties of limits, including: (1) the limit of a constant k is equal to k; (2) the limit of x as x approaches a is equal to a; (3) the limit of kf(x) is equal to k times the limit of f(x); (4) the limit of the sum of two functions is equal to the sum of their individual limits. It also provides examples of calculating limits using these properties, such as finding the limit of 7x - 4 as x approaches 2.
Lesson 8: Derivatives of Logarithmic and Exponential Functions (worksheet sol...Matthew Leingang
This document provides solutions to derivatives of exponential, logarithmic, and other functions. It includes:
1) The derivatives of functions such as y=e^2x, y=6^x, y=ln(x^3 + 9), and y=log_3(e^x).
2) Using logarithmic differentiation to find the derivatives of functions like y=x^x^2-1 and y=(x-1)(x-2)(x-3).
3) Taking the derivative of functions involving logarithms, exponents, and square roots such as y=sin^2(x)+2sin(x) and y=x(x-1)^3/
The document presents a linear programming problem to maximize return from investing Rs. 1,00,000 in two stock portfolios. The maximum investment allowed in each portfolio is Rs. 75,000. Portfolio 1 has a 10% average return and risk rating of 4, while Portfolio 2 has 20% return and risk rating of 9. The objective is to maximize return Z = 0.10X1 + 0.20X2, subject to the constraints that total investment cannot exceed Rs. 1,00,000, individual investments are at most Rs. 75,000 each, risk rating must be below 6, and average return must be at least 12%. The optimal solution is to invest Rs. 60,000 in Port
1) The document provides definitions and formulas for calculating derivatives, including the derivative of a function at a point, the derivative function, and common derivatives of basic functions like polynomials, exponentials, logarithms, trigonometric functions, and inverse trigonometric functions.
2) Examples are given of calculating derivatives using the definition of the derivative, for functions like f(x)=3x^2, f(x)=x+1, and f(x)=1/x.
3) Rules are listed for calculating the derivatives of sums, products, quotients of functions using properties of derivatives.
1) An antiderivative of a function f(x) is any function F(x) whose derivative is equal to f(x).
2) The general antiderivative of a function f(x) is written as F(x) + C, where F(x) is a particular antiderivative and C is an arbitrary constant.
3) The indefinite integral notation ∫f(x)dx represents the entire family of antiderivatives for a function f(x), since each value of C defines a different antiderivative.
This document discusses various integral formulas for simple functions including:
1) Potential integrals from a to x of the form ∫f(x)dx
2) Logarithmic integrals of the form ∫f(x)dx/x
3) Exponential integrals of the form ∫e^f(x)dx
4) Trigonometric integrals involving sine, cosine, and tangent functions.
This document summarizes the solution to an exercise with three parts:
1) Part (a) finds the probability density function f(x) of a random variable X based on its integral from -infinity to infinity being 1. It determines that f(x) = 2 and a = 2.
2) Part (b) calculates the expected value E(x) of X by integrating x*f(x) from 0 to 1. It determines the expected value is 1/3.
3) Part (c) calculates the variance V(X) of X by finding its expected value E(X2) and subtracting the square of its expected value. It determines the variance is 1/
The document describes the midpoint circle algorithm for drawing circles on a pixel screen. It explains how the algorithm determines the midpoint between the next two possible consecutive pixels and checks if the midpoint is inside or outside the circle to determine which pixel to illuminate. It provides the mathematical equations and steps used to iteratively calculate the x and y coordinates of each pixel on the circle. The algorithm is implemented in a C++ program to draw a circle on a graphics screen.
The document is a lesson on continuity and infinite limits. It defines infinite limits, including limits approaching positive or negative infinity. It provides examples of evaluating limits at points where a function is not continuous. It also outlines several rules of thumb for manipulating infinite limits, such as the sum or product of an infinite limit with a finite limit being infinite. The document cautions that limits of indeterminate forms like 0×∞ or ∞-∞ require closer examination rather than following rules of thumb. It provides an example of rationalizing an expression to put it in a form where limit laws can be applied.
The document defines:
1) Powers and exponentiation, including rules for multiplying, dividing, and raising powers of numbers.
2) Examples are provided to illustrate the rules.
3) It notes that care must be taken with negative bases and even/odd exponents.
The summary provides the high level definition of powers/exponentiation and notes some key rules and examples are given to illustrate, highlighting the need to consider signs with negative bases. It does not include details of the specific examples or problems shown in the document.
This document provides a methodology for solving definite and indefinite integrals of various types, including simple, logarithmic, exponential, trigonometric, and their inverses. It contains over 40 examples of integrals worked out step-by-step, covering the basic rules for evaluating indefinite integrals of functions like polynomials, trigonometric functions, exponentials, and their inverses.
1. The document provides examples of limits calculations and concepts.
2. Key steps in limit calculations are presented such as evaluating one-sided limits separately if they differ and using algebraic manipulation to simplify expressions before taking the limit.
3. Problem sets with solutions demonstrate various types of limits, including one-sided limits, limits at infinity, limits of rational functions, and limits of trigonometric functions.
This document contains an instructor's resource manual for limits concepts with examples and problem sets. It includes definitions of limits, worked examples of limit calculations, and limit problems to solve.
This document contains an instructor's resource manual for limits concepts with examples and problem sets. It includes definitions of limits, worked examples of limit calculations, and limit problems to solve.
This document provides the solutions manual for Trigonometry 10th Edition by Larson. It includes solutions for all exercises in Chapter 2 on Analytic Trigonometry. The chapter covers fundamental trigonometric identities, verifying identities, solving trigonometric equations, sum and difference formulas, and multiple-angle and product-to-sum formulas. The solutions provide step-by-step workings to arrive at the answers for each problem.
This document provides the solutions manual for Trigonometry 10th Edition by Larson. It includes solutions for all exercises in Chapter 2 on Analytic Trigonometry. The chapter covers fundamental trigonometric identities, verifying identities, solving trigonometric equations, sum and difference formulas, and multiple-angle and product-to-sum formulas. The solutions provide step-by-step workings to arrive at the answers for each problem.
Trigonometry 10th edition larson solutions manual.
Full download: https://goo.gl/gFVG5A
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This document provides formulas and rules for taking derivatives and integrals of common functions including polynomials, trigonometric functions, inverse trigonometric functions, exponentials, logarithms, and others. It also describes techniques for evaluating integrals using substitution, integration by parts, trigonometric substitutions, partial fractions, and splitting products and quotients of trigonometric functions.
This document provides formulas and definitions for calculus, including derivatives, integrals, exponents, logarithms, trigonometric functions and identities, hyperbolic functions, and other mathematical concepts. It includes over 50 formulas and definitions summarized across two pages with accompanying graphs.
This document provides formulas and definitions for calculus, including:
- Derivatives and integrals of basic functions
- Exponent rules
- Logarithm rules
- Trigonometric functions and their inverses
- Trigonometric identities
- Hyperbolic functions and their inverses
- Graphs of trigonometric and hyperbolic functions
This document provides formulas and definitions for calculus, including derivatives, integrals, exponents, logarithms, trigonometric functions, and hyperbolic functions. It includes formulas for derivatives of sums, products, quotients, and compositions of functions. It also includes trigonometric identities, graphs of trigonometric functions, and definitions of inverse trigonometric functions and hyperbolic functions.
1. This document provides formulas and definitions for calculus, trigonometry, and hyperbolic functions.
2. It includes formulas for limits, derivatives, integrals, exponents, logarithms, trigonometric functions and their inverses, identities, and hyperbolic functions and their inverses.
3. Graphs are also included to illustrate various trigonometric and hyperbolic functions.
1. This document provides formulas and definitions for calculus, trigonometry, and hyperbolic functions.
2. It includes formulas for limits, derivatives, integrals, exponents, logarithms, trigonometric functions and their inverses, identities, and hyperbolic functions and their inverses.
3. Graphs are also included to illustrate various trigonometric and hyperbolic functions.
1. This document provides formulas and definitions for calculus, trigonometry, and hyperbolic functions.
2. It includes formulas for limits, derivatives, integrals, exponents, logarithms, trigonometric functions and their inverses, identities, and hyperbolic functions and their inverses.
3. Graphs are also included to illustrate various trigonometric and hyperbolic functions.
1. This document provides formulas and definitions for calculus, trigonometry, and hyperbolic functions.
2. It includes formulas for limits, derivatives, integrals, exponents, logarithms, trigonometric functions and their inverses, identities, and hyperbolic functions and their inverses.
3. Graphs are also included to illustrate various trigonometric and hyperbolic functions.
1. This document provides formulas and definitions for calculus, trigonometry, and hyperbolic functions.
2. It includes formulas for limits, derivatives, integrals, exponents, logarithms, trigonometric functions and their inverses, identities, and hyperbolic functions and their inverses.
3. Graphs are also included to illustrate various trigonometric and hyperbolic functions.
1. This document provides formulas and definitions for calculus, trigonometry, and hyperbolic functions.
2. It includes formulas for limits, derivatives, integrals, exponents, logarithms, trigonometric functions and their inverses, identities, and hyperbolic functions and their inverses.
3. Graphs are also included to illustrate various trigonometric and hyperbolic functions.
1. This document provides formulas and definitions for calculus, trigonometry, and hyperbolic functions. It includes over 50 formulas across various topics of calculus and trigonometry arranged in tables.
2. Graphs are also included showing the relationships between trigonometric, inverse trigonometric, hyperbolic, and inverse hyperbolic functions.
3. Trigonometric identities are defined relating trigonometric functions of sums and differences of angles.
1. This document provides formulas and definitions for calculus, trigonometry, and hyperbolic functions.
2. It includes formulas for limits, derivatives, integrals, exponents, logarithms, trigonometric functions and their inverses, identities, and hyperbolic functions and their inverses.
3. Graphs are also included to illustrate various trigonometric and hyperbolic functions.
Formulario de Calculo Diferencial-IntegralErick Chevez
This document provides a 3-sentence summary of the key information:
The document is a formula sheet for calculus that includes definitions and formulas for derivatives, integrals, logarithms, exponents, trigonometric functions and their inverses, hyperbolic functions, and other calculus topics. It also contains 5 graphs illustrating trigonometric, inverse trigonometric, and hyperbolic functions. The document is in Spanish and provides contact information for the author as well as links to related websites.
This document provides an overview of wound healing, its functions, stages, mechanisms, factors affecting it, and complications.
A wound is a break in the integrity of the skin or tissues, which may be associated with disruption of the structure and function.
Healing is the body’s response to injury in an attempt to restore normal structure and functions.
Healing can occur in two ways: Regeneration and Repair
There are 4 phases of wound healing: hemostasis, inflammation, proliferation, and remodeling. This document also describes the mechanism of wound healing. Factors that affect healing include infection, uncontrolled diabetes, poor nutrition, age, anemia, the presence of foreign bodies, etc.
Complications of wound healing like infection, hyperpigmentation of scar, contractures, and keloid formation.
This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
Leveraging Generative AI to Drive Nonprofit InnovationTechSoup
In this webinar, participants learned how to utilize Generative AI to streamline operations and elevate member engagement. Amazon Web Service experts provided a customer specific use cases and dived into low/no-code tools that are quick and easy to deploy through Amazon Web Service (AWS.)
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
Exploiting Artificial Intelligence for Empowering Researchers and Faculty,
International FDP on Fundamentals of Research in Social Sciences
at Integral University, Lucknow, 06.06.2024
By Dr. Vinod Kumar Kanvaria
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UPRAHUL
This Dissertation explores the particular circumstances of Mirzapur, a region located in the
core of India. Mirzapur, with its varied terrains and abundant biodiversity, offers an optimal
environment for investigating the changes in vegetation cover dynamics. Our study utilizes
advanced technologies such as GIS (Geographic Information Systems) and Remote sensing to
analyze the transformations that have taken place over the course of a decade.
The complex relationship between human activities and the environment has been the focus
of extensive research and worry. As the global community grapples with swift urbanization,
population expansion, and economic progress, the effects on natural ecosystems are becoming
more evident. A crucial element of this impact is the alteration of vegetation cover, which plays a
significant role in maintaining the ecological equilibrium of our planet.Land serves as the foundation for all human activities and provides the necessary materials for
these activities. As the most crucial natural resource, its utilization by humans results in different
'Land uses,' which are determined by both human activities and the physical characteristics of the
land.
The utilization of land is impacted by human needs and environmental factors. In countries
like India, rapid population growth and the emphasis on extensive resource exploitation can lead
to significant land degradation, adversely affecting the region's land cover.
Therefore, human intervention has significantly influenced land use patterns over many
centuries, evolving its structure over time and space. In the present era, these changes have
accelerated due to factors such as agriculture and urbanization. Information regarding land use and
cover is essential for various planning and management tasks related to the Earth's surface,
providing crucial environmental data for scientific, resource management, policy purposes, and
diverse human activities.
Accurate understanding of land use and cover is imperative for the development planning
of any area. Consequently, a wide range of professionals, including earth system scientists, land
and water managers, and urban planners, are interested in obtaining data on land use and cover
changes, conversion trends, and other related patterns. The spatial dimensions of land use and
cover support policymakers and scientists in making well-informed decisions, as alterations in
these patterns indicate shifts in economic and social conditions. Monitoring such changes with the
help of Advanced technologies like Remote Sensing and Geographic Information Systems is
crucial for coordinated efforts across different administrative levels. Advanced technologies like
Remote Sensing and Geographic Information Systems
9
Changes in vegetation cover refer to variations in the distribution, composition, and overall
structure of plant communities across different temporal and spatial scales. These changes can
occur natural.
বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...
Main Java[All of the Base Concepts}.docxadhitya5119
This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
1. Limites Trigonométricos Resolvidos
Sete páginas e 34 limites resolvidos
1
Usar o limite fundamental e alguns artifícios : 1lim
0
=
→ x
senx
x
1.
x
x
x sen
lim
0→
= ? à
x
x
x sen
lim
0→
=
0
0
, é uma indeterminação.
x
x
x sen
lim
0→
=
x
xx sen
1
lim
0→
=
x
x
x
sen
lim
1
0→
= 1 logo
x
x
x sen
lim
0→
= 1
2.
x
x
x
4sen
lim
0→
= ? à
x
x
x
4sen
lim
0→
=
0
0
à
x
x
x 4
4sen
.4lim
0→
= 4.
y
y
y
sen
lim
0→
=4.1= 4 logo
x
x
x
4sen
lim
0→
=4
3.
x
x
x 2
5sen
lim
0→
= ? à =
→ x
x
x 5
5sen
.
2
5
lim
0
=
→ y
y
y
sen
.
2
5
lim
0 2
5
logo
x
x
x 2
5sen
lim
0→
=
2
5
4.
nx
mx
x
sen
lim
0→
= ? à
nx
mx
x
sen
lim
0→
=
mx
mx
n
m
x
sen
.lim
0→
=
n
m
.
y
y
y
sen
lim
0→
=
n
m
.1=
n
m
logo
nx
mx
x
sen
lim
0→
=
n
m
5.
x
x
x 2sen
3sen
lim
0→
= ? à
x
x
x 2sen
3sen
lim
0→
= =
→
x
x
x
x
x 2sen
3sen
lim
0
=
→
x
x
x
x
x
2
2sen
.2
3
3sen
.3
lim
0
.
2
3
2
2sen
lim
3
3sen
lim
0
0
=
→
→
x
x
x
x
x
x
. 1.
2
3
sen
lim
sen
lim
0
0
=
→
→
t
t
y
y
t
y
=
2
3
logo
x
x
x 2sen
3sen
lim
0→
=
2
3
6.
sennx
senmx
x 0
lim
→
= ? à
nx
mx
x sen
sen
lim
0→
=
x
nx
x
mx
x sen
sen
lim
0→
=
nx
nx
n
mx
mx
m
x sen
.
sen
.
lim
0→
=
nx
nx
mx
mx
n
m
x sen
sen
.lim
0→
=
n
m
Logo
sennx
senmx
x 0
lim
→
=
n
m
7. =
→ x
tgx
x 0
lim ? à =
→ x
tgx
x 0
lim
0
0
à =
→ x
tgx
x 0
lim =
→ x
x
x
x
cos
sen
lim
0
=
→ xx
x
x
1
.
cos
sen
lim
0
xx
x
x cos
1
.
sen
lim
0→
=
xx
x
xx cos
1
lim.
sen
lim
00 →→
= 1 Logo =
→ x
tgx
x 0
lim 1
8.
( )
1
1
lim 2
2
1 −
−
→ a
atg
a
= ? à
( )
1
1
lim 2
2
1 −
−
→ a
atg
a
=
0
0
àFazendo
→
→
−=
0
1
,12
t
x
at à
( )
t
ttg
t 0
lim
→
=1
logo
( )
1
1
lim 2
2
1 −
−
→ a
atg
a
=1
2. Limites Trigonométricos Resolvidos
Sete páginas e 34 limites resolvidos
2
9.
xx
xx
x 2sen
3sen
lim
0 +
−
→
= ? à
xx
xx
x 2sen
3sen
lim
0 +
−
→
=
0
0
à ( )
xx
xx
xf
2sen
3sen
+
−
= =
+
−
x
x
x
x
x
x
5sen
1.
3sen
1.
=
+
−
x
x
x
x
x
x
.5
5sen
.51.
.3
3sen
.31.
=
x
x
x
x
.5
5sen
.51
.3
3sen
.31
+
−
à
0
lim
→x
x
x
x
x
.5
5sen
.51
.3
3sen
.31
+
−
=
51
31
+
−
=
6
2−
=
3
1
− logo
xx
xx
x 2sen
3sen
lim
0 +
−
→
=
3
1
−
10. 30
sen
lim
x
xtgx
x
−
→
= ? à 30
sen
lim
x
xtgx
x
−
→
=
xx
x
xx
x
x cos1
1
.
sen
.
cos
1
.
sen
lim 2
2
0 +→
=
2
1
( ) 3
sen
x
xtgx
xf
−
= = 3
sen
cos
sen
x
x
x
x
−
= 3
cos
cos.sensen
x
x
xxx −
=
( )
xx
xx
cos.
cos1.sen
3
−
=
x
x
xx
x
cos
cos1
.
1
.
sen
2
−
=
x
x
x
x
xx
x
cos1
cos1
.
cos
cos1
.
1
.
sen
2 +
+−
=
xx
x
xx
x
cos1
1
.
cos1
.
cos
1
.
sen
2
2
+
−
=
xx
x
xx
x
cos1
1
.
sen
.
cos
1
.
sen
2
2
+
Logo 30
sen
lim
x
xtgx
x
−
→
=
2
1
11. 30
sen11
lim
x
xtgx
x
+−+
→
=? à
xtgxx
xtgx
x sen11
1
.
sen
lim 30 +++
−
→
=
xtgxxx
x
xx
x
x sen11
1
.
cos1
1
.
sen
.
cos
1
.
sen
lim 2
2
0 ++++→
=
2
1
.
2
1
.
1
1
.
1
1
.1 =
4
1
( ) 3
11
x
senxtgx
xf
+−+
= =
xtgxx
xtgx
sen11
1
.
sen11
3
+++
−−+
=
xtgxx
xtgx
sen11
1
.
sen
3
+++
−
30
sen11
lim
x
xtgx
x
+−+
→
=
4
1
12.
ax
ax
ax −
−
→
sensen
lim = ? à
ax
ax
ax −
−
→
sensen
lim =
−
+
−
→
2
.2
2
cos.
2
sen2
lim
ax
axax
ax
=
1
2
cos.
.
2
.2
)
2
sen(2
lim
+
−
−
→
ax
ax
ax
ax
= acos Logo
ax
ax
ax −
−
→
sensen
lim = cosa
3. Limites Trigonométricos Resolvidos
Sete páginas e 34 limites resolvidos
3
13.
( )
a
xax
a
sensen
lim
0
−+
→
= ? à
( )
a
xax
a
sensen
lim
0
−+
→
=
1
2
cos.
.
2
.2
2
sen2
lim
++
−
−+
→
xax
ax
xax
aa
=
1
2
2
cos.
.
2
.2
2
sen2
lim
+
→
ax
a
a
aa
= xcos Logo
( )
a
xax
a
sensen
lim
0
−+
→
=cosx
14.
( )
a
xax
a
coscos
lim
0
−+
→
= ? à
( )
a
xax
a
coscos
lim
0
−+
→
=
a
xaxxax
a
−−
++
−
→
2
sen.
2
sen2
lim
0
=
−
−
+
−
→
2
.2
2
sen.
2
2
sen.2
lim
0 a
aax
a
=
−
−
+
−
→
2
2
sen
.
2
2
senlim
0 a
a
ax
a
= xsen− Logo
( )
a
xax
a
coscos
lim
0
−+
→
=-senx
15.
ax
ax
ax −
−
→
secsec
lim = ? à
ax
ax
ax −
−
→
secsec
lim =
ax
ax
ax −
−
→
cos
1
cos
1
lim =
ax
ax
xa
ax −
−
→
cos.cos
coscos
lim =
( ) axax
xa
ax cos.cos.
coscos
lim
−
−
→
=
( ) axax
xaxa
ax cos.cos.
2
sen.
2
sen.2
lim
−
−
+
−
→
=
axxa
xaxa
ax cos.cos
1
.
2
.2
2
sen
.
1
2
sen.2
lim
−
−
−
+
−
→
=
axxa
xaxa
ax cos.cos
1
.
2
2
sen
.
1
2
sen
lim
−
−
+
→
=
aa
a
cos.cos
1
.1.
1
sen
=
aa
a
cos
1
.
cos
sen
= atga sec. Logo
ax
ax
ax −
−
→
secsec
lim = atga sec.
16.
x
x
x sec1
lim
2
0 −→
= ? à
x
x
x sec1
lim
2
0 −→
=
( )xxx
xx
cos1
1
.
cos
1
.
sen
1
lim
2
20
+
−
→
= 2−
( )
x
x
xf
cos
1
1
2
−
= =
x
x
x
cos
1cos
2
−
=
( )x
xx
cos1.1
cos.2
−−
=
( ) ( )
( )x
x
xx
x
cos1
cos1
.
cos
1
.
cos1
1
2 +
+−
−
=
( )xxx
x
cos1
1
.
cos
1
.
cos1
1
2
2
+
−
−
=
( )xxx
x
cos1
1
.
cos
1
.
sen
1
2
2
+
−
4. Limites Trigonométricos Resolvidos
Sete páginas e 34 limites resolvidos
4
17.
tgx
gx
x −
−
→ 1
cot1
lim
4
π
= ? à
tgx
gx
x −
−
→ 1
cot1
lim
4
π
=
tgx
tgx
x −
−
→ 1
1
1
lim
4
π
=
tgx
tgx
tgx
x −
−
→ 1
1
lim
4
π
=
tgx
tgx
tgx
x −
−−
→ 1
)1.(1
lim
4
π
=
tgxx
1
lim
4
−
→
π
= 1− Logo
tgx
gx
x −
−
→ 1
cot1
lim
4
π
= -1
18.
x
x
x 2
3
0 sen
cos1
lim
−
→
= ? à
x
x
x 2
3
0 sen
cos1
lim
−
→
=
( )( )
x
xxx
x 2
2
0 cos1
coscos1.cos1
lim
−
++−
→
=
( )( )
( )( )xx
xxx
x cos1.cos1
coscos1.cos1
lim
2
0 +−
++−
→
=
x
xx
x cos1
coscos1
lim
2
0 +
++
→
=
2
3
Logo
x
x
x 2
3
0 sen
cos1
lim
−
→
=
2
3
19.
x
x
x cos.21
3sen
lim
3
−→
π
= ? à
x
x
x cos.21
3sen
lim
3
−→
π
=
( )
1
cos.21.sen
lim
3
xx
x
+
−
→
π
= 3−
( )
x
x
xf
cos.21
3sen
−
= =
( )
x
xx
cos.21
2sen
−
+
=
x
xxxx
cos.21
cos.2sen2cos.sen
−
+
=
( )
x
xxxxx
cos.21
cos.cos.sen.21cos2.sen 2
−
+−
=
( )[ ]
x
xxx
cos.21
cos21cos2.sen 22
−
+−
=
[ ]
x
xx
cos.21
1cos4.sen 2
−
−
=
( )( )
x
coxcoxx
cos.21
.21..21.sen
−
+−
− =
( )
1
cos.21.sen xx +
−
20.
tgx
xx
x −
−
→ 1
cossen
lim
4
π
= ? à
tgx
xx
x −
−
→ 1
cossen
lim
4
π
= ( )x
x
coslim
4
−
→π
=
2
2
−
( )
tgx
xx
xf
−
−
=
1
cossen
=
x
x
xx
cos
sen
1
cossen
−
−
=
x
x
xx
cos
sen
1
cossen
−
−
=
x
xx
xx
cos
sencos
cossen
−
−
=
( )
x
xx
xx
cos
cossen.1
cossen
−−
−
=
xx
xxx
sencos
cos
.
1
cossen
−
−
− = xcos−
21. ( ) )sec(cos.3lim
3
xx
x
π−
→
= ? à ( ) )sec(cos.3lim
3
xx
x
π−
→
= ∞.0
( ) ( ) )sec(cos.3 xxxf π−= =( )
( )x
x
πsen
1
.3 − =
( )x
x
ππ −
−
sen
3
=
( )x
x
ππ −
−
3sen
3
=
( )
( )x
x
−
−
3.
3sen.
1
π
πππ
=
( )
( )x
x
ππ
πππ
−
−
3
3sen.
1
à ( ) )sec(cos.3lim
3
xx
x
π−
→
=
( )
( )x
xx
ππ
πππ
−
−→
3
3sen.
1
lim
3
=
π
1
22. )
1
sen(.lim
x
x
x→∝
= ? à )
1
sen(.lim
x
x
x→∝
= 0.∞
x
x
x 1
1
sen
lim
→∝
= 1
sen
lim
0
=
→ t
t
t
à Fazendo
→
+∞→
=
0
1
t
x
x
t
5. Limites Trigonométricos Resolvidos
Sete páginas e 34 limites resolvidos
5
23.
1sen.3sen.2
1sensen.2
lim 2
2
6 +−
−+
→ xx
xx
x π
= ? à
1sen.3sen.2
1sensen.2
lim 2
2
6 +−
−+
→ xx
xx
x π
=
x
x
x sen1
sen1
lim
6
+−
+
→π
=
6
sen1
6
sen1
π
π
+−
+
=
2
1
1
2
1
1
+−
+
= 3− à ( )
1sen.3sen.2
1sensen.2
2
2
+−
−+
=
xx
xx
xf =
( )
( )1sen.
2
1
sen
1sen.
2
1
sen
−
−
+
−
xx
xx
=
( )
( )1sen
1sen
−
+
x
x
=
x
x
sen1
sen1
+−
+
24. ( )
−
→ 2
.1lim
1
x
tgx
x
π
= ? à ( )
−
→ 2
.1lim
1
x
tgx
x
π
= ∞.0 à ( ) ( )
−=
2
.1
x
tgxxf
π
=
( )
−−
22
cot.1
x
gx
ππ
=
( )
−
−
22
1
x
tg
x
ππ
=
( )
−
−
22
2
.1.
2
x
tg
x
ππ
π
π
=
( )x
x
tg
−
−
1.
2
22
2
π
ππ
π =
−
−
22
22
2
x
x
tg
ππ
ππ
π à
( )
−
→ 2
.1lim
1
x
tgx
x
π
=
−
−
→
22
22
2
lim
1
x
x
tg
x
ππ
ππ
π =
( )
t
ttg
t 0
lim
2
→
π =
π
2
Fazendo uma mudança de variável,
temos :
→
→
−=
0
1
2 t
x
x
x
t
ππ
25.
( )x
x
x πsen
1
lim
2
1
−
→
= ? à
( )x
x
x πsen
1
lim
2
1
−
→
=
( )
( )x
x
x
x
ππ
πππ
−
−
+
→ sen.
1
lim
1
=
π
2
( )
x
x
xf
πsen
1 2
−
= =
( )( )
( )x
xx
ππ −
+−
sen
1.1
=
( )
( )x
x
x
−
−
+
1
sen
1
ππ
=
( )
( )x
x
x
−
−
+
1.
sen.
1
π
πππ
=
( )
( )x
x
x
ππ
πππ
−
−
+
sen.
1
26.
−
→
xgxg
x 2
cot.2cotlim
0
π
= ? à
−
→
xgxg
x 2
cot.2cotlim
0
π
= 0.∞
( )
−= xgxgxf
2
cot.2cot
π
= tgxxg .2cot =
xtg
tgx
2
=
xtg
tgx
tgx
2
1
2
−
=
tgx
xtg
tgx
.2
1
.
2
−
=
2
1 2
xtg−
−
→
xgxg
x 2
cot.2cotlim
0
π
=
2
1
lim
2
0
xtg
x
−
→
=
2
1
27.
x
xx
x 2
3
0 sen
coscos
lim
−
→
= 11102
2
1 ...1
lim
tttt
t
t +++++
−
→
=
12
1
−
( )
x
xx
xf 2
3
sen
coscos −
= = 12
23
1 t
tt
−
−
=
( )
( )( )11102
2
...1.1
1.
ttttt
tt
+++++−
−−
= 11102
2
...1 tttt
t
+++++
−
63.2
coscos xxt ==
→
→
1
0
t
x
xt cos6
= , xt 212
cos= , 122
1sen tx −=
6. Limites Trigonométricos Resolvidos
Sete páginas e 34 limites resolvidos
6
BriotxRuffini :
1 0 0 ... 0 -1
1 • 1 1 ... 1 1
1 1 1 ... 1 0
28.
xx
xx
x sencos
12cos2sen
lim
4
−
−−
→π
= ? à
xx
xx
x sencos
12cos2sen
lim
4
−
−−
→π
= ( )x
x
cos.2lim
4
−
→π
=
4
cos.2
π
− =
2
2
.2− =
2−
( )
xx
xx
xf
sencos
12cos2sen
−
−−
= =
( )
xx
xxx
sencos
11cos2cossen.2 2
−
−−−
=
xx
xxx
sencos
11cos2cos.sen.2 2
−
−+−
=
xx
xxx
sencos
cos2cos.sen.2 2
−
−
=
( )
xx
xxx
sencos
sencos.cos.2
−
−−
= xcos.2−
29.
( )
112
1sen
lim
1
−−
−
→
x
x
x
= ? à
( )
112
1sen
lim
1 −−
−
→ x
x
x
=
( )
( ) 1
112
.
1
1sen
.
2
1
lim
1
+−
−
−
→
x
x
x
x
= 1
( ) ( )
112
1sen
−−
−
=
x
x
xf =
( )
112
112
.
112
1sen
+−
+−
−−
−
x
x
x
x
=
( )
1
112
.
112
1sen +−
−−
− x
x
x
=
( )
( ) 1
112
.
1.2
1sen +−
−
− x
x
x
=
( )
( ) 1
112
.
1
1sen
.
2
1 +−
−
− x
x
x
30.
3
cos.21
lim
3
ππ
−
−
→ x
x
x
= ? à
3
cos.21
lim
3
ππ
−
−
→ x
x
x
=
−
−
+
→
2
3
2
3sen
.
2
3sen.2lim
3 x
x
x
x π
π
π
π
=
.
2
33sen.2
+ππ
= .
2
3
2
sen.2
π
= .
3
sen.2
π
= 3
2
3
.2 =
( )
3
cos.21
π
−
−
=
x
x
xf =
3
cos
2
1
.2
π
−
−
x
x
=
3
cos
3
cos.2
π
π
−
−
x
x
=
( )
−
−
−
+
−
2
3.2.1
2
3sen.
2
3sen2.2
x
xx
π
ππ
=
−
−
+
2
3
2
3sen.
2
3sen.2
x
xx
π
ππ
=
−
−
+
2
3
2
3sen
.
2
3sen.2
x
x
x
π
π
π
31.
xx
x
x sen.
2cos1
lim
0
−
→
= ? à
xx
x
x sen.
2cos1
lim
0
−
→
=
x
x
x
sen.2
lim
3
π
→
= 2
7. Limites Trigonométricos Resolvidos
Sete páginas e 34 limites resolvidos
7
( )
xx
x
xf
sen.
2cos1−
= =
( )
xx
x
sen.
sen211 2
−−
=
xx
x
sen.
sen211 2
+−
=
xx
x
sen.
sen.2 2
=
x
xsen.2
32.
xx
x
x sen1sen1
lim
0 −−+→
= ? à
xx
x
x sen1sen1
lim
0 −−+→
=
x
x
xx
x sen.2
sen1sen1
lim
0
−++
→
=
1.2
11+
=1
( )
xx
x
xf
sen1sen1 −−+
= =
( )
( )xx
xxx
sen1sen1
sen1sen1.
−−+
−++
=
( )
xx
xxx
sen1sen1
sen1sen1.
+−+
−++
=
( )
x
xxx
sen.2
sen1sen1. −++
=
x
x
xx
sen
.2
sen1sen1 −++
=
1.2
11+
= 1
33.
xx
x
x sencos
2cos
lim
0 −→
=
1
sencos
lim
0
xx
x
+
→
=
2
2
2
2
+ = 2
( )
xx
x
xf
sencos
2cos
−
= =
( )
( )( )xxxx
xxx
sencos.sencos
sencos.2cos
+−
+
=
( )
xx
xxx
22
sencos
sencos.2cos
−
+
=
( )
x
xxx
2cos
sencos.2cos +
=
( )
x
xxx
2cos
sencos.2cos +
=
1
sencos xx +
=
2
2
2
2
+ = 2
34.
3
sen.23
lim
3
ππ
−
−
→ x
x
x
= ? à
3
sen.23
lim
3
ππ
−
−
→ x
x
x
=
3
sen
2
3
.2
lim
3
ππ
−
−
→ x
x
x
=
3
sen
3
sen.2
lim
3
π
π
π
−
−
→ x
x
x
=
3
2
3cos.
2
3sen.2
lim
3
π
ππ
π
−
+
−
→ x
xx
x
=
3
3
2
3
3
cos.
2
3
3
sen.2
lim
3
π
ππ
π −
+
−
→
x
xx
x
=
( )
3
3.1
6
3
cos.
6
3
sen.2
lim
3
x
xx
x
−−
+
−
→
π
ππ
π
35. ?