Q: Which of the following limits is an example of the indeterminate form $$0 \times \infty$$?

$$\displaystyle \lim_{x \to 0^+} x \ln x$$

Q: Evaluate $$\displaystyle \lim_{t \to \infty} \frac{e$$^{5t}$$}{t^2$}$$ and interpret the end behavior.

The limit is $$\infty$$; as $$t \to \infty$$, the function increases without bound.

Q: Which of the following limits is of the indeterminate form $$1^\infty$$ and requires logarithmic transformation to solve?

$$\displaystyle \lim_{x \to \infty} (1 + 3/x)^x$$

Q: Which of the following is the correct first step to evaluate $$\displaystyle \lim_{\theta \to \frac{\pi}{2}^-} (\tan \theta - \sec \theta)$$?

Rewrite $$\tan \theta$$ and $$\sec \theta in $$terms of $$\sin \theta$$ and $$\cos \theta$$, then combine into a single fraction.

Q: Suppose $$\displaystyle \lim_{x \to c} \frac{f(x)}{g(x)} is of $$the form $$\frac{0}{0}. $$Which of the following must be true to apply L'Hôpital's Rule?

$$f$$ and $$g$$ are differentiable near $$c$$, and $$g'(x) \neq 0$$ near $$c.$$

Q: Which of the following best describes the end behavior of $$y = \frac{x+5}{x-2} as$$ $$x \to \infty$$?

The function approaches $$y=1 as$$ $$x \to \infty.$$

Question 1

Which of the following limits is an example of the indeterminate form $$0 	imes \infty$$?

Accepted Answer

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

Question 2

Evaluate $$\displaystyle \lim_{x 	o 2} \frac{x^2$ + 4x - 12}{x^2$ - 6x + 8}$$ and describe the graph feature at $$x=2.$$

Accepted Answer

Limit is $$-4$$; there is a hole (removable discontinuity) at $$x=2.$$

Question 3

Which of the following is the correct application of L'Hôpital's Rule to $$\displaystyle \lim_{x 	o 1} \frac{\ln x}{3x^2$ - 3}$$?

Accepted Answer

Take derivatives: numerator $$\frac{1}{x}$$, denominator $$6x$$, so limit is $$\frac{1}{6}.$$

Question 4

Given $$\displaystyle \lim_{x 	o \infty} \frac{3x^2$ + x}{2x^2$ - 5}$$, what is the horizontal asymptote of the function?

Accepted Answer

$$y = \frac{3}{2}$$

Question 5

Evaluate $$\displaystyle \lim_{t 	o \infty} \frac{e$$^{5t}$$}{t^2$}$$ and interpret the end behavior.

Accepted Answer

The limit is $$\infty$$; as $$t 	o \infty$$, the function increases without bound.

Question 6

Which is the correct strategy to evaluate $$\displaystyle \lim_{x 	o 0^+} x \ln x$$?

Accepted Answer

Rewrite as $$\displaystyle \frac{\ln x}{1/x}$$ and apply L'Hôpital's Rule.

Question 7

Evaluate $$\displaystyle \lim_{x 	o \infty} x$$^{3/x}$$.

Accepted Answer

1

Question 8

Which of the following limits is of the indeterminate form $$1^\infty$$ and requires logarithmic transformation to solve?

Accepted Answer

$$\displaystyle \lim_{x 	o \infty} (1 + 3/x)^x$$

Question 9

Evaluate $$\displaystyle \lim_{x 	o \infty} (1 + 3/x)^x.$$

Accepted Answer

$$e^3$$

Question 10

Which of the following is the correct first step to evaluate $$\displaystyle \lim_{	heta 	o \frac{\pi}{2}^-} (	an 	heta - \sec 	heta)$$?

Accepted Answer

Rewrite $$	an 	heta$$ and $$\sec 	heta in $$terms of $$\sin 	heta$$ and $$\cos 	heta$$, then combine into a single fraction.

Question 11

For the function $$y = \frac{x+5}{x-2}$$, which of the following statements is true?

Accepted Answer

There is a vertical asymptote at $$x=2$$ and a horizontal asymptote at $$y=1.$$

Question 12

Which of the following is NOT an indeterminate form that L'Hôpital's Rule can be directly applied to?

Accepted Answer

$$0 	imes \infty$$

Question 13

Suppose $$\displaystyle \lim_{x 	o c} \frac{f(x)}{g(x)} is of $$the form $$\frac{0}{0}. $$Which of the following must be true to apply L'Hôpital's Rule?

Accepted Answer

$$f$$ and $$g$$ are differentiable near $$c$$, and $$g'(x) 
eq 0$$ near $$c.$$

Question 14

Which of the following best describes the end behavior of $$y = \frac{x+5}{x-2} as$$ $$x 	o \infty$$?

Accepted Answer

The function approaches $$y=1 as$$ $$x 	o \infty.$$

Question 15

If $$\displaystyle \lim_{x 	o 0} \frac{e^x - 1}{x}$$, what is the value of the limit?

Accepted Answer

1

Question 16

Which of the following is the correct process for evaluating $$\displaystyle \lim_{x 	o 0} x$$^{x}$$?

Accepted Answer

Let $$y = x^x$$, take $$\ln to $$get $$\ln y = x \ln x$$, find $$\lim_{x 	o 0} x \ln x = 0$$, so $$y 	o 1.$$

Question 17

Which of the following is a necessary algebraic step before applying L'Hôpital's Rule to $$\displaystyle \lim_{x 	o 3} \frac{x^2$ - 9}{x - 3}$$?

Accepted Answer

Factor numerator as $$(x-3)(x+3)$$ and cancel $$(x-3).$$

Question 18

Which of the following best describes the concavity of $$y = \frac{x+5}{x-2}$$ for $$x \u003e 2$$?

Accepted Answer

Concave up, since $$\frac{d^2$y}{dx^2$} = \frac{14}{(x-2)^3$} \u003e 0$$ for $$x \u003e 2.$$

Limits Involving Indeterminate Forms and L'Hôpital's Rule

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