Q: Which of the following limits is NOT an indeterminate form suitable for L'Hospital's Rule?

$$\lim_{x \to 0} \frac{2e^x}{x}$$

Q: Which of the following is a correct application of L'Hospital's Rule for $$\lim_{x \to a} \frac{f(x)}{g(x)}$$?

$$\lim_{x \to a} \frac{f'(x)}{g'(x)}$$

Q: If $$\lim_{x \to a} \frac{f(x)}{g(x)}$$ yields $$\frac{\infty}{\infty}$$, what does L'Hospital's Rule allow you to do?

Take the derivatives and compute $$\lim_{x \to a} \frac{f'(x)}{g'(x)}.$$

Question 1

Which of the following is a necessary condition for applying L'Hospital's Rule to evaluate $$\lim_{x 	o a} \frac{f(x)}{g(x)}$$?

Accepted Answer

The limit must yield an indeterminate form such as $$\frac{0}{0} or$$ $$\frac{\infty}{\infty}.$$

Question 2

Evaluate $$\lim_{x 	o 0} \frac{e$$^{-2x}$$ - 1}{x}$$ using L'Hospital's Rule.

Accepted Answer

$$2$$

Question 3

Which of the following limits is NOT an indeterminate form suitable for L'Hospital's Rule?

Accepted Answer

$$\lim_{x 	o 0} \frac{2e^x}{x}$$

Question 4

If $$\lim_{x 	o a} \frac{f(x)}{g(x)}$$ yields $$\frac{0}{0}$$, what is the next step according to L'Hospital's Rule?

Accepted Answer

Take the derivatives of $$f(x)$$ and $$g(x)$$ and compute $$\lim_{x 	o a} \frac{f'(x)}{g'(x)}.$$

Question 5

Evaluate $$\lim_{x 	o 0} x \ln(x)$$ using L'Hospital's Rule.

Accepted Answer

$$0$$

Question 6

Which of the following is a correct application of L'Hospital's Rule for $$\lim_{x 	o a} \frac{f(x)}{g(x)}$$?

Accepted Answer

$$\lim_{x 	o a} \frac{f'(x)}{g'(x)}$$

Question 7

Suppose $$\lim_{x 	o 0} \frac{\ln(x)}{x}$$ yields $$\frac{-\infty}{0}$$, which is not indeterminate. What is the value of the limit?

Accepted Answer

$$-\infty$$

Question 8

A medical researcher models the decay of a drug in the bloodstream as $$C(t) = e$$^{-kt}$$, where k$$ \u003e 0$$. What is $$\lim_{t 	o \infty} C(t)$$?

Accepted Answer

$$0$$

Question 9

Which of the following forms is NOT considered indeterminate for L'Hospital's Rule?

Accepted Answer

$$\frac{1}{\infty}$$

Question 10

Evaluate $$\lim_{x 	o 0} \frac{\sin(x)}{x}$$ using L'Hospital's Rule.

Accepted Answer

$$1$$

Question 11

A chemical reaction rate is modeled by $$R(x) = \frac{e$$^{-x}$$}{x} as$$ $$x 	o 0^+. $$What is the behavior of $$R(x)$$?

Accepted Answer

$$R(x) 	o \infty$$

Question 12

Which step is necessary before applying L'Hospital's Rule to $$\lim_{x 	o 0} x \ln(x)$$?

Accepted Answer

Rewrite $$x \ln(x) as$$ $$\frac{\ln(x)}{1/x} to $$obtain an indeterminate form.

Question 13

If $$\lim_{x 	o a} \frac{f(x)}{g(x)}$$ yields $$\frac{\infty}{\infty}$$, what does L'Hospital's Rule allow you to do?

Accepted Answer

Take the derivatives and compute $$\lim_{x 	o a} \frac{f'(x)}{g'(x)}.$$

Question 14

Evaluate $$\lim_{x 	o 0} \frac{1 - e$$^{-2x}$$}{x}$$ using L'Hospital's Rule.

Accepted Answer

$$2$$

Question 15

A population model is given by $$P(t) = \frac{e$$^{2t}$$}{t} as$$ $$t 	o 0^+. $$What is the limit?

Accepted Answer

$$\infty$$

Question 16

Which of the following is a real-world scenario where L'Hospital's Rule can be used?

Accepted Answer

Calculating the instantaneous rate of change of a drug concentration as it approaches zero.

L'Hospital's Rule and Indeterminate Forms

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