The document provides two examples of limits involving radicals. In the first example, the limit is evaluated as x approaches positive infinity and negative infinity. The limit is 1 as x approaches positive infinity, and -1 as x approaches negative infinity. In the second example, the limit is simplified by dividing and multiplying by the conjugate, and the limit is evaluated to be 0 as x approaches infinity.
We solve limits by rationalizing. This is the second technique you may learn after limits by factoring. We solve two examples step by step.
Watch video: http://www.youtube.com/watch?v=8CtpuojMJzA
More videos and lessons: http://www.intuitive-calculus.com/solving-limits.html
We develop a new method to optimize portfolios of options in a market where European calls and puts are available with many exercise prices for each of several potentially correlated underlying assets. We identify the combination of asset-specific option payoffs that maximizes the Sharpe ratio of the overall portfolio: such payoffs are the unique solution to a system of integral equations, which reduce to a linear matrix equation under suitable representations of the underlying probabilities. Even when implied volatilities are all higher than historical volatilities, it can be optimal to sell options on some assets while buying options on others, as hedging demand outweighs demand for asset-specific returns.
We solve limits by rationalizing. This is the second technique you may learn after limits by factoring. We solve two examples step by step.
Watch video: http://www.youtube.com/watch?v=8CtpuojMJzA
More videos and lessons: http://www.intuitive-calculus.com/solving-limits.html
We develop a new method to optimize portfolios of options in a market where European calls and puts are available with many exercise prices for each of several potentially correlated underlying assets. We identify the combination of asset-specific option payoffs that maximizes the Sharpe ratio of the overall portfolio: such payoffs are the unique solution to a system of integral equations, which reduce to a linear matrix equation under suitable representations of the underlying probabilities. Even when implied volatilities are all higher than historical volatilities, it can be optimal to sell options on some assets while buying options on others, as hedging demand outweighs demand for asset-specific returns.
Integration by substitution is the chain rule in reverse.
NOTE: the final location is section specific. Section 1 (morning) is in SILV 703, Section 11 (afternoon) is in CANT 200
This presentation gives the basic idea about the methods of solving ODEs
The methods like variation of parameters, undetermined coefficient method, 1/f(D) method, Particular integral and complimentary functions of an ODE
2 Dimensional Wave Equation Analytical and Numerical SolutionAmr Mousa
2 Dimensional Wave Equation Analytical and Numerical Solution
This project aims to solve the wave equation on a 2d square plate and simulate the output in an user-friendly MATLAB-GUI
you can find the gui in mathworks file-exchange here
https://www.mathworks.com/matlabcentral/fileexchange/55117-2d-wave-equation-simulation-numerical-solution-gui
I am Eugeny G. I am a Calculus Assignment Expert at mathsassignmenthelp.com. I hold a Master's in Mathematics from, Columbia University. I have been helping students with their assignments for the past 8 years. I solve assignments related to Calculus.
Visit mathsassignmenthelp.com or email info@mathsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Calculus Assignment.
These slides contain information about Euler method,Improved Euler and Runge-kutta's method.How these methods are helpful and applied to our questions are detailed discussed in the slides.
Day 5 of the Intuitive Online Calculus Course: The Squeeze TheoremPablo Antuna
In this presentation we learn about the Squeeze Theorem. We first try to get the intuition behind it, why it must be true. Then we apply it to solve the Fundamental Trigonometric Limit. This limit is very important for solving other trigonometric limits.
To solve this limit we use a little bit of geometry and then apply the Squeeze Theorem.
Integration by substitution is the chain rule in reverse.
NOTE: the final location is section specific. Section 1 (morning) is in SILV 703, Section 11 (afternoon) is in CANT 200
This presentation gives the basic idea about the methods of solving ODEs
The methods like variation of parameters, undetermined coefficient method, 1/f(D) method, Particular integral and complimentary functions of an ODE
2 Dimensional Wave Equation Analytical and Numerical SolutionAmr Mousa
2 Dimensional Wave Equation Analytical and Numerical Solution
This project aims to solve the wave equation on a 2d square plate and simulate the output in an user-friendly MATLAB-GUI
you can find the gui in mathworks file-exchange here
https://www.mathworks.com/matlabcentral/fileexchange/55117-2d-wave-equation-simulation-numerical-solution-gui
I am Eugeny G. I am a Calculus Assignment Expert at mathsassignmenthelp.com. I hold a Master's in Mathematics from, Columbia University. I have been helping students with their assignments for the past 8 years. I solve assignments related to Calculus.
Visit mathsassignmenthelp.com or email info@mathsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Calculus Assignment.
These slides contain information about Euler method,Improved Euler and Runge-kutta's method.How these methods are helpful and applied to our questions are detailed discussed in the slides.
Day 5 of the Intuitive Online Calculus Course: The Squeeze TheoremPablo Antuna
In this presentation we learn about the Squeeze Theorem. We first try to get the intuition behind it, why it must be true. Then we apply it to solve the Fundamental Trigonometric Limit. This limit is very important for solving other trigonometric limits.
To solve this limit we use a little bit of geometry and then apply the Squeeze Theorem.
In this presentation we learn to solve limits using the limit definition of number e.
For more lessons and videos: http://www.intuitive-calculus.com/solving-limits.html
Derivatives of Trigonometric Functions, Part 2Pablo Antuna
In this presentation we find the derivative of cos(x) and then we solve two examples.
For more videos and lessons: http://www.intuitive-calculus.com/der...
In this presentation we introduce the chain rule and we solve two basic examples explaining each of the steps.
For more lessons: http://www.intuitive-calculus.com/chain-rule.html
In this presentation we solve two more examples of implicit differentiation problems. We use a faster, more direct method.
For more lessons visit: http://www.intuitive-calculus.com/implicit-differentiation.html
In this video we learn how to solve limits that involve trigonometric functions. It is all based on using the fundamental trigonometric limit, which is proved using the squeeze theorem.
For more lessons: http://www.intuitive-calculus.com/solving-limits.html
Watch video: http://www.youtube.com/watch?v=1RqXMJWcRIA
In this video we talk about the LIATE rule, this is the secret for choosing u and v' correctly. Then we learn another useful trick for integration by parts.
Watch video: http://www.youtube.com/edit?ns=1&video_id=1SEUGvHTcQ0
For more lessons: http://www.intuitive-calculus.com/integration-by-parts.html
In this video we learn how to solve limits by factoring and cancelling. This is one of the most simple and powerful techniques for solving limits.
Watch video: http://www.youtube.com/watch?v=r0Qw5gZuTYE
For more videos and lessons: http://www.intuitive-calculus.com/solving-limits.html
Day 3 of Free Intuitive Calculus Course: Limits by FactoringPablo Antuna
Today we focus on limits by factoring. We solve limits by factoring and cancelling. This is one of the basic techniques for solving limits. We talk about the idea behind this technique and we solve some examples step by step.
Integrals by Trigonometric SubstitutionPablo Antuna
In this video we learn the basics of trigonometric substitution and why it works. We talk about all the basic cases of integrals you can solve using trigonometric substitution.
For more lessons and presentations:
Mat 121-Limits education tutorial 22 I.pdfyavig57063
limitsExample: A function C=f(d) gives the number of classes
C, a student takes in a day, d of the week. What does
f(Monday)=4 mean?
Solution. From f(Monday)=4, we see that the input day
is Monday while the output value, number of courses is
4. Thus, the student takes 4 classes on Mondays.Function: is a rule which assigns an element in
the domain to an element in the range in such a
way that each element in the domain
corresponds to exactly one element in the range.
The notation f(x) read “f of x” or “f at x” means
function of x while the notion y=f(x) means y is a
function of x. The letter x represents the input
value, or independent variable
Limit & Continuity of Functions - Differential Calculus by Arun Umraossuserd6b1fd
This books explains about limits and continuity and is base for derivative calculus. Suitable for CBSE Class XII students who are preparing for IIT JEE.
7. Limits with Radicals
Example 1
Let’s consider the limit:
lim
x→∞
√
x2 + 1
x + 1
Here we have a radical.
We’re going to do something similar to our basic technique.
8. Limits with Radicals
Example 1
Let’s consider the limit:
lim
x→∞
√
x2 + 1
x + 1
Here we have a radical.
We’re going to do something similar to our basic technique.
So, we divide and multiply by x2 inside the radical symbol:
9. Limits with Radicals
Example 1
Let’s consider the limit:
lim
x→∞
√
x2 + 1
x + 1
Here we have a radical.
We’re going to do something similar to our basic technique.
So, we divide and multiply by x2 inside the radical symbol:
lim
x→∞
x2+1
x2 .x2
x + 1
10. Limits with Radicals
Example 1
Let’s consider the limit:
lim
x→∞
√
x2 + 1
x + 1
Here we have a radical.
We’re going to do something similar to our basic technique.
So, we divide and multiply by x2 inside the radical symbol:
lim
x→∞
x2+1
x2 .x2
x + 1
lim
x→∞
1 + 1
x2 x2
x + 1
15. Limits with Radicals
Example 1
lim
x→∞
1 + 1
x2 x2
x + 1
Here we have two cases to consider:
1. x < 0.
As we are taking a positive square root:
16. Limits with Radicals
Example 1
lim
x→∞
1 + 1
x2 x2
x + 1
Here we have two cases to consider:
1. x < 0.
As we are taking a positive square root:
lim
x→∞
(−x)
1 + 1
x2
x + 1
17. Limits with Radicals
Example 1
lim
x→∞
1 + 1
x2 x2
x + 1
Here we have two cases to consider:
1. x < 0.
As we are taking a positive square root:
lim
x→∞
(−x)
1 + 1
x2
x + 1
We can also write this as:
18. Limits with Radicals
Example 1
lim
x→∞
1 + 1
x2 x2
x + 1
Here we have two cases to consider:
1. x < 0.
As we are taking a positive square root:
lim
x→∞
(−x)
1 + 1
x2
x + 1
We can also write this as:
− lim
x→∞
1 + 1
x2
x+1
x
22. Limits with Radicals
Example 1
− lim
x→∞
1 + 1
x2
x+1
x
= − lim
x→∞
1 + 1
x2
1 + 1
x
Now, as we take the limit when x → +∞:
23. Limits with Radicals
Example 1
− lim
x→∞
1 + 1
x2
x+1
x
= − lim
x→∞
1 + 1
x2
1 + 1
x
Now, as we take the limit when x → +∞:
= − lim
x→∞
1 + 1
x2
1 + 1
x
24. Limits with Radicals
Example 1
− lim
x→∞
1 + 1
x2
x+1
x
= − lim
x→∞
1 + 1
x2
1 + 1
x
Now, as we take the limit when x → +∞:
= − lim
x→∞
1 +
U
0
1
x2
1 + 1
x
25. Limits with Radicals
Example 1
− lim
x→∞
1 + 1
x2
x+1
x
= − lim
x→∞
1 + 1
x2
1 + 1
x
Now, as we take the limit when x → +∞:
= − lim
x→∞
1 +
U
0
1
x2
1 +
¡
¡!
0
1
x
26. Limits with Radicals
Example 1
− lim
x→∞
1 + 1
x2
x+1
x
= − lim
x→∞
1 + 1
x2
1 + 1
x
Now, as we take the limit when x → +∞:
= − lim
x→∞
1 +
U
0
1
x2
1 +
¡
¡!
0
1
x
= −
√
1
1
= −1
27. Limits with Radicals
Example 1
− lim
x→∞
1 + 1
x2
x+1
x
= − lim
x→∞
1 + 1
x2
1 + 1
x
Now, as we take the limit when x → +∞:
= − lim
x→∞
1 +
U
0
1
x2
1 +
¡
¡!
0
1
x
= −
√
1
1
= −1
32. Limits with Radicals
Example 1
Now the second case:
2. x 0.
lim
x→∞
1 + 1
x2 x2
x + 1
We take the positive square root:
33. Limits with Radicals
Example 1
Now the second case:
2. x 0.
lim
x→∞
1 + 1
x2 x2
x + 1
We take the positive square root:
lim
x→∞
(x)
1 + 1
x2
x + 1
34. Limits with Radicals
Example 1
Now the second case:
2. x 0.
lim
x→∞
1 + 1
x2 x2
x + 1
We take the positive square root:
lim
x→∞
(x)
1 + 1
x2
x + 1
= lim
x→∞
1 + 1
x2
x+1
x
35. Limits with Radicals
Example 1
Now the second case:
2. x 0.
lim
x→∞
1 + 1
x2 x2
x + 1
We take the positive square root:
lim
x→∞
(x)
1 + 1
x2
x + 1
= lim
x→∞
1 + 1
x2
x+1
x
= lim
x→∞
1 + 1
x2
1 + 1
x
36. Limits with Radicals
Example 1
Now the second case:
2. x 0.
lim
x→∞
1 + 1
x2 x2
x + 1
We take the positive square root:
lim
x→∞
(x)
1 + 1
x2
x + 1
= lim
x→∞
1 + 1
x2
x+1
x
= lim
x→∞
1 +
U
0
1
x2
1 + 1
x
37. Limits with Radicals
Example 1
Now the second case:
2. x 0.
lim
x→∞
1 + 1
x2 x2
x + 1
We take the positive square root:
lim
x→∞
(x)
1 + 1
x2
x + 1
= lim
x→∞
1 + 1
x2
x+1
x
= lim
x→∞
1 +
U
0
1
x2
1 +
¡
¡!
0
1
x
38. Limits with Radicals
Example 1
Now the second case:
2. x 0.
lim
x→∞
1 + 1
x2 x2
x + 1
We take the positive square root:
lim
x→∞
(x)
1 + 1
x2
x + 1
= lim
x→∞
1 + 1
x2
x+1
x
= lim
x→∞
1 +
U
0
1
x2
1 +
¡
¡!
0
1
x
= 1
39. Limits with Radicals
Example 1
Now the second case:
2. x 0.
lim
x→∞
1 + 1
x2 x2
x + 1
We take the positive square root:
lim
x→∞
(x)
1 + 1
x2
x + 1
= lim
x→∞
1 + 1
x2
x+1
x
= lim
x→∞
1 +
U
0
1
x2
1 +
¡
¡!
0
1
x
= 1
48. Limits with Radicals
Example 2
Let’s now consider the limit:
lim
x→∞
x2 + 1 − x2 − 1
This limit is different, but easily solved.
49. Limits with Radicals
Example 2
Let’s now consider the limit:
lim
x→∞
x2 + 1 − x2 − 1
This limit is different, but easily solved.
We just need to divide and multiply by the conjugate:
50. Limits with Radicals
Example 2
Let’s now consider the limit:
lim
x→∞
x2 + 1 − x2 − 1
This limit is different, but easily solved.
We just need to divide and multiply by the conjugate:
lim
x→∞
x2 + 1 − x2 − 1 .
√
x2 + 1 +
√
x2 − 1
√
x2 + 1 +
√
x2 − 1
51. Limits with Radicals
Example 2
Let’s now consider the limit:
lim
x→∞
x2 + 1 − x2 − 1
This limit is different, but easily solved.
We just need to divide and multiply by the conjugate:
lim
x→∞
x2 + 1 − x2 − 1 .
√
x2 + 1 +
√
x2 − 1
√
x2 + 1 +
√
x2 − 1
Doing the algebra in the numerator:
52. Limits with Radicals
Example 2
Let’s now consider the limit:
lim
x→∞
x2 + 1 − x2 − 1
This limit is different, but easily solved.
We just need to divide and multiply by the conjugate:
lim
x→∞
x2 + 1 − x2 − 1 .
√
x2 + 1 +
√
x2 − 1
√
x2 + 1 +
√
x2 − 1
Doing the algebra in the numerator:
lim
x→∞
√
x2 + 1
2
−
√
x2 − 1
2
√
x2 + 1 +
√
x2 − 1