Q: Suppose you encounter the limit $$\displaystyle \lim_{x \to \infty} \frac{\ln x}{x}. $$Which of the following is the correct application of L'Hôpital's Rule?

$$\displaystyle \lim_{x \to \infty} \frac{1/x}{1} = 0$$

Q: Which of the following limits leads to the indeterminate form $$1^\infty$$?

$$\displaystyle \lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x$$

Q: Which of the following limits is determinate (not indeterminate)?

$$\displaystyle \lim_{x \to 0} \frac{1}{x}$$

Q: Which of the following is a correct step in applying L'Hôpital's Rule to $$\displaystyle \lim_{x \to 0} \frac{\ln(1+x)}{x}$$?

$$\displaystyle \lim_{x \to 0} \frac{1/(1+x)}{1}$$

Question 1

Which of the following is an indeterminate form that can be evaluated using L'Hôpital's Rule?

Accepted Answer

$$\displaystyle \frac{0}{0}$$

Question 2

Evaluate the following limit using L'Hôpital's Rule: $$\displaystyle \lim_{x 	o 0} \frac{e$$^{2x}$$ - 1}{x}$$

Accepted Answer

$$2$$

Question 3

Which theorem justifies the use of L'Hôpital's Rule for differentiable functions on an interval?

Accepted Answer

The Extended Mean Value Theorem

Question 4

Suppose you encounter the limit $$\displaystyle \lim_{x 	o \infty} \frac{\ln x}{x}. $$Which of the following is the correct application of L'Hôpital's Rule?

Accepted Answer

$$\displaystyle \lim_{x 	o \infty} \frac{1/x}{1} = 0$$

Question 5

Which of the following limits leads to the indeterminate form $$1^\infty$$?

Accepted Answer

$$\displaystyle \lim_{x 	o \infty} \left(1 + \frac{1}{x}ight)^x$$

Question 6

A chemical reaction rate is modeled by $$R(x) = \frac{e$$^{2x}$$ - 1}{x}$$, where $$x is $$the concentration of a reactant. What is the instantaneous rate as $$x$$ approaches zero?

Accepted Answer

$$2$$

Question 7

Which of the following is NOT an indeterminate form?

Accepted Answer

$$\displaystyle \frac{1}{0}$$

Question 8

Evaluate $$\displaystyle \lim_{x 	o 1^+} \frac{x - 1 - \ln x}{(x - 1) \ln x}$$ using L'Hôpital's Rule.

Accepted Answer

$$\frac{1}{2}$$

Question 9

A supersonic jet's drag force is modeled by $$F(x) = \frac{3x^2$ - 1}{2x^2$ + 1} as$$ $$x 	o \infty. $$What is the limiting value of the drag force?

Accepted Answer

$$\frac{3}{2}$$

Question 10

Which of the following statements about L'Hôpital's Rule is correct?

Accepted Answer

It can only be applied to limits that result in $$\displaystyle \frac{0}{0} or$$ $$\displaystyle \frac{\infty}{\infty}$$ forms.

Question 11

Evaluate $$\displaystyle \lim_{x 	o 0^+} x^x.$$

Accepted Answer

$$1$$

Question 12

A furnace's temperature change is modeled by $$T(x) = \frac{2x^2$ - 2}{x + 1} as$$ $$x 	o -1. $$What is the limit?

Accepted Answer

$$-4$$

Question 13

Which of the following limits is determinate (not indeterminate)?

Accepted Answer

$$\displaystyle \lim_{x 	o 0} \frac{1}{x}$$

Question 14

Evaluate $$\displaystyle \lim_{x 	o \infty} \left(1 + \frac{1}{x}ight)^x.$$

Accepted Answer

$$e$$

Question 15

Which of the following is a correct step in applying L'Hôpital's Rule to $$\displaystyle \lim_{x 	o 0} \frac{\ln(1+x)}{x}$$?

Accepted Answer

$$\displaystyle \lim_{x 	o 0} \frac{1/(1+x)}{1}$$

Question 16

A canyon's erosion rate is modeled by $$E(x) = \frac{1}{\ln x} - \frac{1}{x - 1} as$$ $$x 	o 1^+. $$What is the limit?

Accepted Answer

$$\frac{1}{2}$$

Question 17

Which of the following is a necessary condition for applying L'Hôpital's Rule?

Accepted Answer

The limit must result in $$\displaystyle \frac{0}{0} or$$ $$\displaystyle \frac{\infty}{\infty}.$$

Question 18

Evaluate $$\displaystyle \lim_{x 	o 0} x \cdot \ln x.$$

Accepted Answer

$$0$$

Question 19

Which of the following is an example of an indeterminate form that must be converted before applying L'Hôpital's Rule?

Accepted Answer

$$0^0$$

Indeterminate Forms and L'Hôpital's Rule: Calculus Study Guide

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