Q: Which of the following limits can be evaluated directly without L'Hôpital's Rule?

$$\displaystyle \lim_{x \to 0} x e^x$$

Question 1

Which of the following is the correct application of L'Hôpital's Rule to evaluate $$\displaystyle \lim_{x 	o 0} \frac{	an(x^2$)}{\sin^2$ x}$$?

Accepted Answer

Apply L'Hôpital's Rule once, differentiating numerator and denominator with respect to $$x$$, then evaluate the new limit.

Question 2

Evaluate $$\displaystyle \lim_{x 	o 0} x^2$ \ln x$$ using an appropriate substitution and L'Hôpital's Rule.

Accepted Answer

The limit is $$0.$$

Question 3

Which of the following integrals is improper and requires careful consideration of convergence?

Accepted Answer

$$\displaystyle \$$int_1$$^2 \frac{1}{\ln x} \, dx$$ because $$\ln x$$ approaches $$0 as$$ $$x 	o 1^+.$$

Question 4

Find the value of $$\displaystyle \$$int_1$$^e x e^x \, dx.$$

Accepted Answer

$$(e^e - e)$$

Question 5

A chemical reaction rate is modeled by $$R(x) = x^2$ \ln x$$ for $$x \u003e 0. $$What happens to the rate as $$x 	o 0^+$$?

Accepted Answer

It approaches $$0.$$

Question 6

Which substitution is most useful for evaluating $$\displaystyle \$$int_1$$^e x e^x \, dx$$?

Accepted Answer

Let $$u = x e^x$$

Question 7

Suppose you are asked to evaluate $$\displaystyle \$$int_1$$^2 \frac{1}{\ln x} \, dx. $$What is the main challenge in computing this integral?

Accepted Answer

The integrand is undefined at $$x=1$$ because $$\ln 1 = 0.$$

Question 8

Which of the following limits can be evaluated directly without L'Hôpital's Rule?

Accepted Answer

$$\displaystyle \lim_{x 	o 0} x e^x$$

Question 9

A medical researcher models the growth of bacteria with $$f(x) = x e^x$$ for $$x \geq 0. $$What is $$\displaystyle \$$int_1$$^e x e^x \, dx$$ physically representing?

Accepted Answer

The total bacteria population from $$x=1 to$$ $$x=e.$$

Question 10

Which of the following is the correct first step in using L'Hôpital's Rule for $$\displaystyle \lim_{x 	o 0} \frac{	an(x^2$)}{\sin^2$ x}$$?

Accepted Answer

Differentiate numerator and denominator with respect to $$x.$$

Question 11

If $$\displaystyle \$$int_1$$^2 \frac{1}{\ln x} \, dx$$ diverges, what is the reason?

Accepted Answer

The integrand is not defined at $$x=1.$$

Question 12

Which of the following is the correct evaluation of $$\displaystyle \lim_{x 	o 0} x^2$ \ln x$$?

Accepted Answer

$$0$$

Question 13

A historian models population growth with $$P(x) = x^2$ \ln x. $$What does $$\displaystyle \lim_{x 	o 0^+} P(x)$$ represent?

Accepted Answer

The initial population is $$0.$$

Question 14

Which of the following is a valid substitution for evaluating $$\displaystyle \$$int_1$$^e x e^x \, dx$$ using integration by parts?

Accepted Answer

Let $$u = x$$, $$dv = e^x dx$$

Question 15

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

Accepted Answer

The integral is improper because $$\ln x is$$ $$0 at$$ $$x=1.$$

Question 16

Which of the following is the correct evaluation of $$\displaystyle \$$int_1$$^e x e^x \, dx$$?

Accepted Answer

$$e^e - e$$

Question 17

A physicist models the energy output of a system as $$E(x) = x^2$ \ln x. $$What is the energy output as $$x 	o 0^+$$?

Accepted Answer

It approaches $$0.$$

Question 18

Which of the following is the correct application of L'Hôpital's Rule to $$\displaystyle \lim_{x 	o 0} \frac{	an(x^2$)}{\sin^2$ x}$$?

Accepted Answer

Differentiate numerator and denominator with respect to $$x$$ and evaluate the new limit.

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

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