L‘Hôspital’s Rule L’ Hopital’s Rule Differential Calculus OPENING QUESTION: Take a moment to reflect on what you already know about calculus and limits. Why might learning how to find the limit of indeterminate forms become beneficial? Monica MasonWhat is L’hopital’s Rule (A video tutorial) : www.khanacademy.org/math/calculus/differential-calculus/v/introduction-to-l-hopital-s-rule
Content Area: MathematicsGrade Level: 12thSummary: The purpose of this instructional PowerPoint is for students tobe able to find the limit of functions in the indeterminate form byapplying L’Hopital’s rule.Learning Objective: By the end of this lesson students will be able to findthe limit of functions with an indeterminate form by applying L’Hopital’srule with 100% accuracy.Content Standard: Recognize and use mathematical ideas and processesthat arise in different settings, with an emphasis on formulating aproblem in mathematical terms, interpreting solutions, mathematicalideas, and the communication of solution strategies.Accomplishment: Students will successfully show a full understanding ofthis activity once they are able to take functions in different forms, adjustthem if necessary to apply L’Hopital’s rule to find the limit of functions inindeterminate form.
L’Hôpital’s Rule• His name is firmly associated with LHôpitals rule for calculating limits involving indeterminate forms 0/0 and ∞/∞. Definition of Indeterminate Limits: www.mathworld.wolfram.com/Indeterminate.html
Example of When the Rule is Needed We see that as X approaches 0 on the numerator it equals zero. We see that the same occurs with the denominator. This leaves us the in determinant form ofHere is where we will apply LHôpitals rule. We will take the derivative of thenumerator and the derivative of the denominator to determine the limits of thisequations. Once we take the derivative of both the numerator and the denominatorwe will see that the limit iswww.Tutorial.math.lamar.edu/Classes.CalcI/LHospitalsRule.aspx(Explanation of L’Hopital’s Rule).
Problem # 1To start this problem, it is important for students to look and see ifthis equation is in an indeterminate form. If the equation is in anindeterminate form apply L’Hopital’s rule. Look at the numeratorand denominator differently. Take the derivative of each and seewhat the limit is as t approaches 1.
Solution to Problem #1In this case we have a 0/0 indeterminate form and if we werereally good at factoring we could factor the numerator anddenominator, simplify and take the limit. However, that’s goingto be more work than just using L’Hopital’s Rule.After taking the derivative of both the numerator and thedenominator (step 1) we will substitute t for 1( step 2) since thisis where we are seeing where the limit approaches as t goes to 1.Simplify (Step 4).
Problem # 2To start this problem, it is important for students to look and see if this equationis in an indeterminate form. If the equation is in an indeterminate form applyL’Hopital’s rule. Look at the numerator and denominator differently. Take thederivative of each and see what the limit is as x approaches infinity. (Hint: Thederivative of the numerator is itself).
Solution to Problem # 2 As we examine this function we know that it’s the indeterminate form so let’s apply L’Hopital’s Rule (Step 1). Now we have a small problem. This new limit is also a indeterminate form (Step 2). However, it’s not really a problem. We know how to deal with these kinds of limits. Just apply L’Hopital’s Rule (Step 3). Sometimes we will need to apply L’Hopital’s Rule more than once.
Problem # 3To start this problem, it is important for students to look and see if thisequation is in an indeterminate form. If the equation is in an indeterminateform apply L’Hopital’s rule. (Hint: Since this equation is not in fraction formyou must manipulate the function to put into a fraction to apply L’Hopital’sRule).
Solution to Problem # 3 At this step we will write the function as a quotient Applying L’Hopital’s rule on this quotient leads to an endless cycle where a limit cannot be determined. Moving the other function (exponential function) to the denominator the limit can be determined using L’Hopital’s rule. Simplify the function using L’Hopital’s rule to find the limit.
Problem # 4To start this problem, it is important for students to look at thisfunction and see that they should spend their time looking at thenatural log of this function.
Solution to Problem # 4 Define this function Now we simplify to apply L’Hopital’s rule and solve. Take the natural log of both sides of the equation.We know that if this was the function how to find the limitusing L’hopital’s rule.So we will rewrite the limit as the function above.
Conclusion• In all during this assignment, students were able to analyze different functions to determine if the function was in the indeterminate form. If the function was in an indeterminate form, students should have manipulated the equation in order to apply L’Hopital’s rule. By using L’Hopital’s rule students were able to find limits of function in the indeterminate form.
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