Prerequisite knowledge

• How to find limits numerically, graphically, and symbolically

• How to find derivatives of polynomial, trigonometric, and exponential functions

• How to write a linear approximation to a curve at a given point

Equipment needed

• Graphing calculator

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.

Graphically, the limit appears to be _______________.

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

x	h(x)

 

 

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

I think the limit is ___________________.

 

 

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

 

 

 

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

 

 

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

 

 

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

The limit is ____________________ .

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

Repeat Part 1 for the function h(x) =

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

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

 

 

2. Graph the numerator and denominator functions separately. Compare their slopes as x grows infinitely large.

 

 

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

 

 

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

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 .

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

1. lim_{x® 0} f(x) = 0, lim_{x® 0} g(x) = 0, and lim_{x® 0} h(x) = 0
2. lim_{x® 0} f(x) = 0, lim_{x® 0} g(x) = 0, and lim_{x® 0} h(x) = 2

 

3. lim_{x® 0} f(x) = 0, lim_{x® 0} g(x) = 0, and lim_{x® 0} h(x) = ¥
4. lim_{x® 0} f(x) = ¥ , lim_{x® 0} g(x) = ¥ , and lim_{x® 0} h(x) = 0

 

5. lim_{x® 0} f(x) = ¥ , lim_{x® 0} g(x) = ¥ , and lim_{x® 0} h(x) = 2
6. lim_{x® 0} f(x) = ¥ , lim_{x® 0} g(x) = ¥ , and lim_{x® 0} h(x) = ¥

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. lim_{x® a} h(x) = _____ b) lim_{x® d} h(x) = _____ c) lim_{x® ¥} h(x) = ____

d) lim_{x® b} k(x) = _____ e) lim_{x® d} k(x) = _____ f) lim_{x® ¥} k(x) = ____

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

1. lim_{x® 0} = ______
2. lim_{x® 0} = ______
3. lim_{x® 0+} = ______
4. lim_{x® = ______}
5. lim_{x® ¥} = ______
6. lim_{x® 0} = ______
7. lim_{x® ¥}

L’Hôpital’s Rule states: ________________________________________________________________________________________________________________________________________________

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

lim_{x® 0} = lim_{x® 0}

= lim_{x® 0}

= ½

Let f(x) =

1. Explain why some graphs of f(x) may give false information about lim_{x® 0} f(x). (Hint: Try the window [-1,1] by [-.05, 1].)

 

 

2. Explain why tables may give false information about lim_{x® 0} f(x). (Hint: Try tables with increments of 0.01.)

 

 

 

3. Use L’Hôpital’s Rule to find lim_{x® 0} f(x). Show your work.

 

 

 

 

 

lim_{x® 0} f(x) = ______

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. 5^th 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.