Q: Which of the following limits is of the indeterminate form $$\frac{\infty}{\infty}$$?

$$\lim_{x \to \infty} \frac{\ln x}{x}$$

Q: Which of the following is a valid application of L'Hôpital's Rule?

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

Question 1

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

Accepted Answer

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

Question 2

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

Accepted Answer

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

Question 3

Which of the following limits is of the indeterminate form $$\frac{\infty}{\infty}$$?

Accepted Answer

$$\lim_{x 	o \infty} \frac{\ln x}{x}$$

Question 4

Suppose you are evaluating $$\lim_{x 	o 0} \frac{\sin(2x)}{20x}. $$Which of the following is the correct value?

Accepted Answer

$$\frac{1}{10}$$

Question 5

Which of the following is NOT an indeterminate form?

Accepted Answer

$$1^1$$

Question 6

Evaluate the following limit: 
$$
\lim_{x 	o \infty} \frac{e^x}{x^2$}
$$

Accepted Answer

$$\infty$$

Question 7

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

Accepted Answer

Both $$f(x)$$ and $$g(x)$$ must be differentiable near $$x=a.$$

Question 8

Evaluate the following limit using L'Hôpital's Rule: 
$$
\lim_{x 	o 0} \frac{\ln(1+x)}{x}
$$

Accepted Answer

$$1$$

Question 9

Which of the following limits requires algebraic manipulation before applying L'Hôpital's Rule?

Accepted Answer

$$\lim_{x 	o 0} x \ln x$$

Question 10

What is the value of the following limit? 
$$
\lim_{x 	o \infty} \frac{\ln x}{x}
$$

Accepted Answer

$$0$$

Question 11

Which of the following is a valid application of L'Hôpital's Rule?

Accepted Answer

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

Question 12

Evaluate the following limit: 
$$
\lim_{x 	o 0^+} x^x
$$

Accepted Answer

$$0$$

Question 13

Which of the following limits is of the indeterminate form $$0^0$$?

Accepted Answer

$$\lim_{x 	o 0^+} x^x$$

Question 14

Which of the following is NOT a step in applying L'Hôpital's Rule?

Accepted Answer

Multiply the numerator and denominator.

Question 15

Evaluate the following limit using L'Hôpital's Rule: 
$$
\lim_{x 	o 1} \frac{x - 1}{\ln x}
$$

Accepted Answer

$$1$$

Question 16

Which of the following limits is of the indeterminate form $$\$$infty^0$$?

Accepted Answer

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

Question 17

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

Accepted Answer

Calculating the instantaneous rate of change of a chemical reaction when both reactant and product concentrations approach zero.

Question 18

Suppose $$f(x)$$ and $$g(x)$$ are differentiable and $$\lim_{x 	o a} f(x) = 0$$ and $$\lim_{x 	o a} g(x) = 0. $$Which of the following is true about $$\lim_{x 	o a} \frac{f(x)}{g(x)}$$?

Accepted Answer

It is an indeterminate form and may require L'Hôpital's Rule.

Question 19

Which of the following limits can be evaluated using L'Hôpital's Rule? 
(i) $$\lim_{x 	o 0} \frac{\sin x}{x} 
(ii$$) $$\lim_{x 	o 0} \frac{e^x - 1}{x} 
($$iii) $$\lim_{x 	o 0} \frac{1}{x}$$

Accepted Answer

(i) and (ii)

Question 20

Which of the following is a common mistake when applying L'Hôpital's Rule?

Accepted Answer

Applying the rule when the limit is not of the form $$\frac{0}{0} or$$ $$\frac{\infty}{\infty}.$$

L'Hôpital's Rule and Indeterminate Forms: Study Notes

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