Q: Suppose $$\lim_{x \to a} f(x) = 0$$ and $$\lim_{x \to a} g(x) = 0. $$What is the next step to evaluate $$\lim_{x \to a} \frac{f(x)}{g(x)}$$ using L'Hospital's Rule?

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

Q: Which of the following is a correct application of L'Hospital's Rule?

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

Q: Which of the following limits requires two applications of L'Hospital's Rule to evaluate?

$$\lim_{x \to 0} \frac{1 - \cos x}{x^2$}$$

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

Both $$f(x)$$ and $$g(x)$$ must be differentiable near $$x = a$$, and $$\lim_{x 	o a} \frac{f(x)}{g(x)}$$ must yield an indeterminate form $$\frac{0}{0} or$$ $$\frac{\infty}{\infty}.$$

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

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

Question 4

Suppose $$\lim_{x 	o a} f(x) = 0$$ and $$\lim_{x 	o a} g(x) = 0. $$What is the next step to evaluate $$\lim_{x 	o a} \frac{f(x)}{g(x)}$$ using L'Hospital's Rule?

Accepted Answer

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

Question 5

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

Accepted Answer

$$0$$

Question 6

Which of the following is a correct application of L'Hospital's Rule?

Accepted Answer

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

Question 7

If $$\lim_{x 	o a} \frac{f(x)}{g(x)}$$ yields $$\frac{0}{0}$$, but $$\lim_{x 	o a} \frac{f'(x)}{g'(x)}$$ also yields $$\frac{0}{0}$$, what should you do?

Accepted Answer

Apply L'Hospital's Rule again to $$\frac{f'(x)}{g'(x)}.$$

Question 8

Evaluate $$\lim_{x 	o 0} \frac{\sin x}{x}.$$

Accepted Answer

$$1$$

Question 9

Which of the following is an indeterminate form that may require algebraic manipulation before applying L'Hospital's Rule?

Accepted Answer

$$0 \cdot \infty$$

Question 10

A hospital is studying the rate at which a medication leaves the bloodstream. The concentration $$C(t) (in mg/L) at $$time $$t (in $$hours) is modeled by $$C(t) = e$$^{-kt}$$, where k$$ \u003e 0$$. What is $$\lim_{t 	o \infty} C(t)$$?

Accepted Answer

$$0$$

Question 11

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

Accepted Answer

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

Question 12

Which of the following limits requires two applications of L'Hospital's Rule to evaluate?

Accepted Answer

$$\lim_{x 	o 0} \frac{1 - \cos x}{x^2$}$$

Question 13

Evaluate $$\lim_{x 	o \infty} \frac{x}{e^x}$$ using L'Hospital's Rule.

Accepted Answer

$$0$$

Question 14

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

Accepted Answer

Calculating the instantaneous rate of change of a population when both the population and time approach zero.

L'Hospital's Rule and Indeterminate Forms

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