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

$$\lim_{x \to 0} \frac{\sin x}{x}$$

Q: Suppose $$\lim_{x \to a} \frac{f(x)}{g(x)}$$ yields the indeterminate form $$\frac{0}{0}. $$According to L'Hôpital's Rule, which limit should you evaluate?

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

Q: Which of the following limits requires more than one application of L'Hôpital's Rule to evaluate?

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

Question 1

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

Accepted Answer

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

Question 2

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

Accepted Answer

$$1$$

Question 3

Suppose $$\lim_{x 	o a} \frac{f(x)}{g(x)}$$ yields the indeterminate form $$\frac{0}{0}. $$According to L'Hôpital's Rule, which limit should you evaluate?

Accepted Answer

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

Question 4

Evaluate $$\lim_{x 	o \infty} \frac{e^x}{x^2$}$$ using L'Hôpital's Rule.

Accepted Answer

$$\infty$$

Question 5

A hospital is monitoring the concentration of a drug in a patient's bloodstream, modeled by $$C(t) = \frac{e^t}{t^2$}$$, where $$t is $$time in hours. What happens to the concentration as $$t 	o \infty$$?

Accepted Answer

It increases without bound

Question 6

Evaluate $$\lim_{x 	o 1} \frac{\ln x}{x - 1}$$ using L'Hôpital's Rule.

Accepted Answer

$$1$$

Question 7

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

Accepted Answer

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

Question 8

A chemical reaction rate is modeled by $$R(x) = \frac{x^2$ - 16}{x - 4}. $$What is $$\lim_{x 	o 4} R(x)$$?

Accepted Answer

$$8$$

Question 9

Evaluate $$\lim_{x 	o \infty} \frac{4x^2$ - 5x + 7}{1 - 3x^2$}$$ using L'Hôpital's Rule.

Accepted Answer

$$\frac{4}{3}$$

Question 10

Which step is necessary before applying L'Hôpital's Rule to $$\lim_{x 	o a} \frac{f(x)}{g(x)}$$?

Accepted Answer

Verify that the limit yields an indeterminate form $$\frac{0}{0} or$$ $$\frac{\infty}{\infty}$$

Question 11

A medical researcher is analyzing the limit $$\lim_{x 	o 0} \frac{\ln(1 + x)}{x}. $$What is the value of this limit?

Accepted Answer

$$1$$

Question 12

If $$\lim_{x 	o a} \frac{f(x)}{g(x)}$$ after applying L'Hôpital's Rule still gives an indeterminate form, what should you do?

Accepted Answer

Apply L'Hôpital's Rule again

Question 13

Evaluate $$\lim_{x 	o 0^+} \frac{\ln x}{x}$$ using L'Hôpital's Rule.

Accepted Answer

$$-\infty$$

Question 14

Which of the following real-world scenarios could be modeled by an indeterminate form suitable for L'Hôpital's Rule?

Accepted Answer

Determining the limit of a chemical concentration as time approaches infinity when both numerator and denominator grow without bound

Question 15

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

Accepted Answer

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

Question 16

A biologist is studying the growth of bacteria modeled by $$B(x) = \frac{e$$^{2x}$$}{e^x}. $$What is $$\lim_{x 	o \infty} B(x)$$?

Accepted Answer

$$\infty$$

Question 17

Which of the following limits requires more than one application of L'Hôpital's Rule to evaluate?

Accepted Answer

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

Question 18

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

Accepted Answer

$$1$$

Question 19

A physicist is analyzing the limit $$\lim_{x 	o 0} \frac{	an x}{x}. $$What is the value of this limit?

Accepted Answer

$$1$$

L'Hôpital's Rule and Indeterminate Forms 4.7

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