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

When evaluating $$\lim\limits_{x \to a} \frac{f(x)}{g(x)}$$ where both $$f(x)$$ and $$g(x)$$ approach $$0 or$$ $$\infty as$$ $$x \to a$$

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

$$\lim\limits_{x \to 0} \frac{\tan(x^2$)}{\sin^2$ x}$$

Question 1

Evaluate the following limit using L'Hôpital's Rule: $$\lim\limits_{x 	o 0} \frac{	an(x^2$)}{\sin^2$ x}.$$

Accepted Answer

$$0$$

Question 2

Find $$\lim\limits_{x 	o 0^+} x^2$ \ln x.$$

Accepted Answer

$$0$$

Question 3

Evaluate the improper integral $$\$$int_1$$^2 \frac{1}{\ln x} \, dx.$$

Accepted Answer

$$\operatorname{Li}(2) - \operatorname{Li}(1)$$, where $$\operatorname{Li}(x) is $$the logarithmic integral

Question 4

Compute $$\$$int_1$$^2 x^e \, dx.$$

Accepted Answer

$$\frac{2$$^{e+1}$$ - 1$$^{e+1}$$}{e+1}$$

Question 5

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

Accepted Answer

When evaluating $$\lim\limits_{x 	o a} \frac{f(x)}{g(x)}$$ where both $$f(x)$$ and $$g(x)$$ approach $$0 or$$ $$\infty as$$ $$x 	o a$$

Question 6

Suppose a chemical reaction rate is modeled by $$R(x) = x^2$ \ln x$$ for $$x \u003e 0. $$What happens to the reaction rate as the concentration $$x$$ approaches zero?

Accepted Answer

The reaction rate approaches $$0$$

Question 7

Evaluate $$\lim\limits_{x 	o 0} \frac{\sin(x^2$)}{x^2$}.$$

Accepted Answer

$$1$$

Question 8

A medical dosage is modeled by $$D(x) = \frac{1}{\ln x}$$ for $$x \u003e 1. $$What is the behavior of $$D(x) as$$ $$x$$ approaches $$1$$ from the right?

Accepted Answer

$$D(x)$$ approaches $$\infty$$

Question 9

Which of the following integrals is improper?

Accepted Answer

$$\$$int_1$$^2 \frac{1}{\ln x} \, dx$$

Question 10

If $$f(x) = 	an(x^2$)$$ and $$g(x) = \sin^2$ x$$, what is $$\lim\limits_{x 	o 0} \frac{f(x)}{g(x)}$$?

Accepted Answer

$$0$$

Question 11

A population model is given by $$P(x) = x^2$ \ln x$$ for $$x \u003e 0. $$What is the instantaneous rate of change of the population at $$x=1$$?

Accepted Answer

$$0$$

Question 12

Which of the following statements about $$\$$int_1$$^2 \frac{1}{\ln x} \, dx is $$true?

Accepted Answer

It is an improper integral because $$\ln x$$ approaches $$0 as$$ $$x 	o 1$$

Question 13

Find the derivative of $$f(x) = x^e.$$

Accepted Answer

$$e x$$^{e-1}$$

Question 14

Which of the following limits requires L'Hôpital's Rule to evaluate?

Accepted Answer

$$\lim\limits_{x 	o 0} \frac{	an(x^2$)}{\sin^2$ x}$$

Question 15

A historical event's rate of change is modeled by $$R(x) = x^2$ \ln x. $$What is $$R'(x)$$?

Accepted Answer

$$2x \ln x + x$$

Calculus Study Notes: Limits, L'Hôpital's Rule, and Integration

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