I’ve Reached My Limit!

 

Students have already studied procedures to take the limit, including direct substitution, cancellation, rationalization, and graphing or using table values. Some functions have limits that cannot be found using standard algebraic techniques. The objective of this lab is to lead students to the discovery of L’Hôpital’s Rule. First, students examine functions that approach the indeterminant form as x approaches a specific value. By comparing the limits numerically and graphically, they determine the value of the limit at x. Next, students examine functions that approach the indeterminant form as x approaches a specific value.

Prerequisite knowledge

Equipment needed

Possible answers given

Part 1

Consider the function h(x) = . Investigate how this function behaves near zero.

  1. Determine the limit as x approaches zero by zooming in on it graphically.

    2. Graphically, the limit appears to be between .1702 and .1631_.

     

     

     

     

     

     

     

     

    b) Determine the limit as x approaches zero by zooming in on it numerically. Make a table to record your investigations.

    x

    h(x)

     

    x

    H(x)

     

     

     

    -.003

    .16717

     

     

     

    -.002

    .167

     

     

     

    -.001

    .16683

     

     

     

    0

    Error

     

     

     

    .001

    .1665

     

     

     

    .002

    .16633

     

     

     

    .003

    .16171

     

     

     

    .004

    .16617
  2. Based on your results in parts a and b, what do you think the limit is?

    3. I think the limit is __.16666666(repeating 6).

  3. Find the linear approximations for the numerator and denominator functions at x = 0.

    4. (The students would already have discussed this technique and its use.)

    For the numerator-

    For the denominator-

  4. Replace the original numerator and denominator with these linearizations and find the limit.

    5. limx® 0 = limx® 0 = .

  5. Compare this limit to the answer from 1c.

    6. They are the same value.

  6. Determine the derivatives of the numerator and the denominator. What is the limit of the ratio of these expressions as x approaches zero?

    7. f’(x) = limx® 0 = limx® 0 =

    g’(x) = 2e2x

     

     

     

     

  7. Compare this limit to the answers from 1c and 1e. Explain.

The values are the same. Functions f(x) and g(x) in the numerator and denominator, respectively, are both continuous and differentiable at x = 0; therefore, they may be approximated using their tangent lines. The limits along these approximations will be the same as the limits of the functions since they are equal at their points of tangency. The numerator and denominator may be replaced by their linear approximations and the limit taken as x approaches 0. The linear approximations are of the form

y = m(x2-x1). The ratio of these approximations, evaluated at x = 0, is the ratio of the slopes. Using the ratio of the derivatives gives the same value as the ratio of the linear approximations.

 

 

Part 2

Repeat Part 1 for the function h(x) = .

Note: If the ( ) are entered incorrectly, the ln gets squared. The intention is, and students may enter on the calculator as, ln((x+1)^2). I think #1 and #2 lead to

wonderful calculator entry discussions as to students' intention.

 

Student answers should be similar to those for Part 1. The limit is 2. If students do not get this limit, a brief discussion on calculator entries with parentheses may be useful.

 

Part 3

Consider the function h(x) = again, this time as x becomes infinitely large.

This function gives students a second indeterminant form .

  1. Determine the limit both graphically and numerically as x grows infinitely large.

    2.  

     

    Graphically, the function appears to be asymptotic to the x-axis. Numerically, the limit approaches 0.

  2. Graph the numerator f(x) and denominator g(x) functions separately and label appropriately. Compare their slopes as x grows infinitely large.

    3. The numerator function has a decreasing slope that seems to approach 0, while the denominator has a constant slope of 1.

  3. Determine the derivatives of the numerator and the denominator functions. What is the ratio of these expressions?

    4. f’(x) = g’(x)= 1

    limx® ¥ = 0

     

  4. Explain the connection between your answers to 3a and 3c.

Both parts have the same value. The numerator becomes almost insignificant in comparison with the denominator. Their ratio gives the ratio of the actual values as x becomes infinitely large.

Part 4

Describe some ways to get around the difficulty of trying to determine a limit of a rational function that takes an indeterminate form such as or .

The limits may be found graphically or numerically as x approaches the specified value. Also, by considering the ratio of the derivatives of the numerator and denominator, the limit of this ratio is the same as that of the original rational function.

Part 5

Create functions of the form h(x) = that satisfy the given requirements.

a) limx® 0 f(x) = 0, limx® 0 g(x) = 0, and limx® 0 h(x) = 0

  1. limx® 0 f(x) = 0, limx® 0 g(x) = 0, and limx® 0 h(x) = 2
  2. limx® 0 f(x) = 0, limx® 0 g(x) = 0, and limx® 0 h(x) = ¥
  3. limx® 0 f(x) = ¥ , limx® 0 g(x) = ¥ , and limx® 0 h(x) = 0
  4. limx® 0 f(x) = ¥ , limx® 0 g(x) = ¥ , and limx® 0 h(x) = 2
  5. limx® 0 f(x) = ¥ , limx® 0 g(x) = ¥ , and limx® 0 h(x) = ¥

Student answers will vary. Students may need to be encouraged to use relatively simple functions to create these new functions.

 

Part 6

Consider the graphs of the functions below.

 

 

Suppose h(x) = and k(x) = . Determine the following limits and explain how you found them. If it is not possible to find the limit, explain why not.

  1. limx® a h(x) = ____ b) limx® d h(x) = ____ c) limx® ¥ h(x)= ____

d) limx® b k(x) = ____ e) limx® d k(x) = ____ f) limx® ¥ k(x) = ____

 

Answers:

  1. limx® a h(x) = ¥ b) limx® d h(x) » -0.179 c) limx® ¥ h(x) = -0.1

d) limx® b k(x) =0 e) limx® d k(x) » -5.6 f) limx® ¥ k(x) » -10

Part 7

Based upon your findings, now try these problems. Be prepared to explain your results graph, numerically, analytically, or verbally.

  1. limx® 0 = ______
  2. limx® 0 = ______
  3. limx® 0+ = ______
  4. limx® = ______
  5. limx® ¥ = ______
  6. limx® 0 = ______
  7. limx® ¥

= ______

 

L’Hôpital’s Rule states: Suppose that f(a) = g(a)=0, that f and g are differentiable on an open interval I containing a, and that g’(x)0 on I if x a. Then

limx® a limx® a .

Find the error in the following incorrect application of L’Hôpital’s Rule.

limx® 0 = limx® 0

= limx® 0

= ½

 

f(0) = g(0) = 0 and both functions are differentiable on an open interval containing 0.

g’(x)0 on I if x 0. The derivatives of the numerator and denominator functions are taken separately until a non-indeterminant form is obtained. In this case, the derivative was taken an extra, unnecessary time before the limit was evaluated.

 

 

Let f(x) = . (Note: Just the x valued is raised to the power, not the cosine.)

  1. Explain why some graphs of f(x) may give false information about limx® 0 f(x).

    2. (Hint: Try the window [-1,1] by [-.05, 1].

  2. Explain why tables may give false information about limx® 0 f(x).

    3. (Hint: Try tables with increments of 0.01.)

  3. Use L’Hôpital’s Rule to find limx® 0 f(x).

    4. (Final answer) limx® 0 = ½

     

     

  4. This is an example of a function for which graphing calculators do not have enough precision to give reliable information. Based on parts a and b explain this statement in your own words.

 

 

 

 

 

Sources:

Finney, Demana, Waits, Kennedy. Calculus – Graphical, Numerical Algebraic. Glenview, IL: Prentice Hall, 2003.

Kamischke, Ellen. A Watched Cup Never Cools. Emeryville, CA: Key Curriculum Press, 1999.

Edwards and Penney. Single Variable Calculus with Analytic Geometry. 5th ed. Upper Saddle River, NJ:Prentice Hall, 1998.

Hilbert, Maceli, Robinson, Schwartz, Seltzer.Calculus – An Active Approach with Projects. New York: John Wiley & Sons, Inc., 1994.