Tractrix A person moves from the origin along the positive axis pulling a weight at the

Numerical and Graphical Analysis In Exercises 14, complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to support your result. lim x0 sin 5x sin 2x

Numerical and Graphical Analysis In Exercises 14, complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to support your result. lim x0 1  ex x

Numerical and Graphical Analysis In Exercises 14, complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to support your result. lim x x5 ex 100 l

Numerical and Graphical Analysis In Exercises 14, complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to support your result. lim x 6x 3x2  2x l

In Exercises 510, evaluate the limit (a) using techniques from Chapters 1 and 3 and (b) using LHpitals Rule. lim x3 2x  3 x2  9

In Exercises 510, evaluate the limit (a) using techniques from Chapters 1 and 3 and (b) using LHpitals Rule. lim x1 2x2  x  3 x 1 l

In Exercises 510, evaluate the limit (a) using techniques from Chapters 1 and 3 and (b) using LHpitals Rule. lim x3 x 1  2 x  3 l

In Exercises 510, evaluate the limit (a) using techniques from Chapters 1 and 3 and (b) using LHpitals Rule. lim x0 sin 4x 2x

In Exercises 510, evaluate the limit (a) using techniques from Chapters 1 and 3 and (b) using LHpitals Rule. lim x 5x2  3x 1 3x2  5 l

In Exercises 510, evaluate the limit (a) using techniques from Chapters 1 and 3 and (b) using LHpitals Rule. lim x 2x 1 4x2 x lim

In Exercises 1136, evaluate the limit, using LHpitals Rule if necessary. (In Exercise 18, is a positive integer.) lim x2 x2  x  2 x  2

In Exercises 1136, evaluate the limit, using LHpitals Rule if necessary. (In Exercise 18, is a positive integer.) lim x1 x2  x  2 x 1 l

In Exercises 1136, evaluate the limit, using LHpitals Rule if necessary. (In Exercise 18, is a positive integer.) lim x0 4  x2  2 x

In Exercises 1136, evaluate the limit, using LHpitals Rule if necessary. (In Exercise 18, is a positive integer.) lim x2 4  x2 x  2 l

In Exercises 1136, evaluate the limit, using LHpitals Rule if necessary. (In Exercise 18, is a positive integer.) lim x0 ex  1  x x

In Exercises 1136, evaluate the limit, using LHpitals Rule if necessary. (In Exercise 18, is a positive integer.) lim x1 ln x2 x2  1

In Exercises 1136, evaluate the limit, using LHpitals Rule if necessary. (In Exercise 18, is a positive integer.) lim x0 ex  1 x x3 li

In Exercises 1136, evaluate the limit, using LHpitals Rule if necessary. (In Exercise 18, is a positive integer.) lim x0 e x  1 x xn lim

In Exercises 1136, evaluate the limit, using LHpitals Rule if necessary. (In Exercise 18, is a positive integer.) lim x0 sin 2x sin 3x

In Exercises 1136, evaluate the limit, using LHpitals Rule if necessary. (In Exercise 18, is a positive integer.) lim x0 sin ax sin bx

In Exercises 1136, evaluate the limit, using LHpitals Rule if necessary. (In Exercise 18, is a positive integer.) lim x0 arcsin x x

In Exercises 1136, evaluate the limit, using LHpitals Rule if necessary. (In Exercise 18, is a positive integer.) lim x1 arctan x   4 x  1 l

In Exercises 1136, evaluate the limit, using LHpitals Rule if necessary. (In Exercise 18, is a positive integer.) lim x 3x2  2x 1 2x2 3 li

In Exercises 1136, evaluate the limit, using LHpitals Rule if necessary. (In Exercise 18, is a positive integer.) lim x x  1 x2 2x 3 lim

In Exercises 1136, evaluate the limit, using LHpitals Rule if necessary. (In Exercise 18, is a positive integer.) lim x x2 2x 3 x  1 li

In Exercises 1136, evaluate the limit, using LHpitals Rule if necessary. (In Exercise 18, is a positive integer.) lim x x3 x 2 li

In Exercises 1136, evaluate the limit, using LHpitals Rule if necessary. (In Exercise 18, is a positive integer.) lim x x3 ex 2

In Exercises 1136, evaluate the limit, using LHpitals Rule if necessary. (In Exercise 18, is a positive integer.) lim x x2 ex l

In Exercises 1136, evaluate the limit, using LHpitals Rule if necessary. (In Exercise 18, is a positive integer.) lim x x x2 1 lim

In Exercises 1136, evaluate the limit, using LHpitals Rule if necessary. (In Exercise 18, is a positive integer.) lim x x2 x2 1 lim

In Exercises 1136, evaluate the limit, using LHpitals Rule if necessary. (In Exercise 18, is a positive integer.) lim x cos x x

In Exercises 1136, evaluate the limit, using LHpitals Rule if necessary. (In Exercise 18, is a positive integer.) lim x sin x x   l

In Exercises 1136, evaluate the limit, using LHpitals Rule if necessary. (In Exercise 18, is a positive integer.) lim x ln x x2

In Exercises 1136, evaluate the limit, using LHpitals Rule if necessary. (In Exercise 18, is a positive integer.) lim x ln x4 x3 l

In Exercises 1136, evaluate the limit, using LHpitals Rule if necessary. (In Exercise 18, is a positive integer.) lim x ex x2

In Exercises 1136, evaluate the limit, using LHpitals Rule if necessary. (In Exercise 18, is a positive integer.) lim x ex 2 x l

In Exercises 3754, (a) describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using LHpitals Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). lim cot x x x ln x

In Exercises 3754, (a) describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using LHpitals Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). lim x0 x 3 lim cot x x

In Exercises 3754, (a) describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using LHpitals Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). lim xx sin 1 x

In Exercises 3754, (a) describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using LHpitals Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). lim x x tan 1 x l

In Exercises 3754, (a) describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using LHpitals Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). lim x0 x1 x

In Exercises 3754, (a) describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using LHpitals Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). lim x0 e x x2 x lim

In Exercises 3754, (a) describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using LHpitals Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). lim x x1 x

In Exercises 3754, (a) describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using LHpitals Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). lim x 1 1 x x l

In Exercises 3754, (a) describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using LHpitals Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). lim x0 1 x1 x li

In Exercises 3754, (a) describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using LHpitals Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). lim x 1 x1 x lim

In Exercises 3754, (a) describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using LHpitals Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). lim x0 3xx 2 l

In Exercises 3754, (a) describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using LHpitals Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). lim x4 3x  4 x4 li

In Exercises 3754, (a) describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using LHpitals Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). lim x1 ln x x1 l

In Exercises 3754, (a) describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using LHpitals Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). lim x0 cos  2  x x

In Exercises 3754, (a) describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using LHpitals Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). lim x2 8 x2  4  x x  2 l

In Exercises 3754, (a) describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using LHpitals Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). lim x2 1 x2  4  x  1 x2  4  li

In Exercises 3754, (a) describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using LHpitals Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). lim x1 3 ln x  2 x  1 l

In Exercises 3754, (a) describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using LHpitals Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). lim x0 10 x  3 x2 l

In Exercises 5558, use a graphing utility to (a) graph the function and (b) find the required limit (if it exists). lim x3 x  3 ln2x  5

In Exercises 5558, use a graphing utility to (a) graph the function and (b) find the required limit (if it exists). lim x0 sin xx l

In Exercises 5558, use a graphing utility to (a) graph the function and (b) find the required limit (if it exists). lim x x2 5x 2  x lim

In Exercises 5558, use a graphing utility to (a) graph the function and (b) find the required limit (if it exists). lim x x3 e2x

List six different indeterminate forms.

State LHpitals Rule.

Find the differentiable functions and that satisfy the specified condition such that and Explain how you obtained your answers. (Note: There are many correct answers.) (a) (b) (c)

Find differentiable functions and such that and Explain how you obtained your answers. (Note: There are many correct answers.)

Numerical Approach Complete the table to show that eventually overpowers

Numerical Approach Complete the table to show that eventually overpowers

Comparing Functions In Exercises 6570, use LHpitals Rule to determine the comparative rates of increase of the functions and where and lim x x2 e5x

Comparing Functions In Exercises 6570, use LHpitals Rule to determine the comparative rates of increase of the functions and where and lim x x3 e2x l

Comparing Functions In Exercises 6570, use LHpitals Rule to determine the comparative rates of increase of the functions and where and lim x ln x3 x l

Comparing Functions In Exercises 6570, use LHpitals Rule to determine the comparative rates of increase of the functions and where and lim x ln x2 x3 li

Comparing Functions In Exercises 6570, use LHpitals Rule to determine the comparative rates of increase of the functions and where and lim x ln xn xm li

Comparing Functions In Exercises 6570, use LHpitals Rule to determine the comparative rates of increase of the functions and where and lim x xm enx l

In Exercises 7174, find any asymptotes and relative extrema that may exist and use a graphing utility to graph the function. (Hint: Some of the limits required in finding asymptotes have been found in preceding exercises.) y  x x > 0

In Exercises 7174, find any asymptotes and relative extrema that may exist and use a graphing utility to graph the function. (Hint: Some of the limits required in finding asymptotes have been found in preceding exercises.) y  x x > 0

In Exercises 7174, find any asymptotes and relative extrema that may exist and use a graphing utility to graph the function. (Hint: Some of the limits required in finding asymptotes have been found in preceding exercises.) y  2xex

In Exercises 7174, find any asymptotes and relative extrema that may exist and use a graphing utility to graph the function. (Hint: Some of the limits required in finding asymptotes have been found in preceding exercises.) y  ln x x

Think About It In Exercises 7578, LHpitals Rule is used incorrectly. Describe the error lim x0 e2x  1 e x  lim x0 2e2x e x  lim x0 2ex  2

Think About It In Exercises 7578, LHpitals Rule is used incorrectly. Describe the error lim x0 sin x  1 x  lim x0  cos x 1

Think About It In Exercises 7578, LHpitals Rule is used incorrectly. Describe the error lim x x cos 1 x  lim x cos1 x 1 x li

Think About It In Exercises 7578, LHpitals Rule is used incorrectly. Describe the error lim x ex 1 ex  lim x ex ex

Analytical Approach In Exercises 79 and 80, (a) explain why LHpitals Rule cannot be used to find the limit, (b) find the limit analytically, and (c) use a graphing utility to graph the function and approximate the limit from the graph. Compare the result with that in part (b). lim x x x2 1  1

Analytical Approach In Exercises 79 and 80, (a) explain why LHpitals Rule cannot be used to find the limit, (b) find the limit analytically, and (c) use a graphing utility to graph the function and approximate the limit from the graph. Compare the result with that in part (b). lim x 2 tan x sec x

Graphical Analysis In Exercises 81 and 82, graph and near What do you notice about these ratios as How does this illustrate LHpitals Rule? fx  sin 3x, gx  sin 4x x

Graphical Analysis In Exercises 81 and 82, graph and near What do you notice about these ratios as How does this illustrate LHpitals Rule? fx  e gx  x 3x

Velocity in a Resisting Medium The velocity of an object falling through a resisting medium such as air or water is given by where is the initial velocity, is the time in seconds, and is the resistance constant of the medium. Use LHpitals Rule to find the formula for the velocity of a falling body in a vacuum by fixing and and letting approach zero. (Assume that the downward direction is positive.)

Compound Interest The formula for the amount in a savings account compounded times per year for years at an interest rate and an initial deposit of is given by Use LHpitals Rule to show that the limiting formula as the number of compoundings per year becomes infinite is given by

The Gamma Function The Gamma Function is defined in terms of the integral of the function given by Show that for any fixed value of the limit of as approaches infinity is zero.

Tractrix A person moves from the origin along the positive axis pulling a weight at the end of a 12-meter rope (see figure). Initially, the weight is located at the point (a) Show that the slope of the tangent line of the path of the weight is (b) Use the result of part (a) to find the equation of the path of the weight. Use a graphing utility to graph the path and compare it with the figure. (c) Find any vertical asymptotes of the graph in part (b). (d) When the person has reached the point how far has the weight moved?

In Exercises 8790, apply the Extended Mean Value Theorem to the functions and on the given interval. Find all values in the interval such that gx  x 0, 1 fx

In Exercises 8790, apply the Extended Mean Value Theorem to the functions and on the given interval. Find all values in the interval such that gx  x 1, 2 fx

In Exercises 8790, apply the Extended Mean Value Theorem to the functions and on the given interval. Find all values in the interval such that fx  sin x, gx  cos x g

In Exercises 8790, apply the Extended Mean Value Theorem to the functions and on the given interval. Find all values in the interval such that fx  ln x,

True or False? In Exercises 9194, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

True or False? In Exercises 9194, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If then

True or False? In Exercises 9194, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If is a polynomial, then

True or False? In Exercises 9194, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If then

Area Find the limit, as approaches 0, of the ratio of the area of the triangle to the total shaded area in the figure.

In Section 1.3, a geometric argument (see figure) was used to prove that (a) Write the area of in terms of (b) Write the area of the shaded region in terms of (c) Write the ratio of the area of to that of the shaded region. (d) Find

Continuous Functions In Exercises 97 and 98, find the value of that makes the function continuous at fx   4x  2 sin 2x 2x3 , c, x  0 x

Continuous Functions In Exercises 97 and 98, find the value of that makes the function continuous at fx   ex x1 x , c, x  0 x  0 fx

Find the values of and such that

Show that for any integer

(a) Let be continuous. Show that (b) Explain the result of part (a) graphically.

Let be continuous. Show that

Sketch the graph of and determine

Use a graphing utility to graph for 0.1, and 0.01. Then evaluate the limit

Consider the limit (a) Describe the type of indeterminate form that is obtained by direct substitution. (b) Evaluate the limit. (c) Use a graphing utility to verify the result of part (b). FOR FURTHER INFORMATION For a geometric approach to this exercise, see the article A Geometric Proof of by John H. Mathews in the College Mathematics Journal.

Prove that if and then

Prove that if and then

Prove the following generalization of the Mean Value Theorem. If is twice differentiable on the closed interval then

Indeterminate Forms Show that the indeterminate forms and do not always have a value of 1 by evaluating each limit. (a) (b) (c)

Calculus History In LHpitals 1696 calculus textbook, he illustrated his rule using the limit of the function as approaches Find this limit.

Consider the function (a) Use a graphing utility to graph the function. Then use the zoom and trace features to investigate (b) Find analytically by writing (c) Can you use LHpitals Rule to find Explain your reasoning.

Evaluate where