$$\lim\limits_{x \to \infty} f(x) = L$$

Question 1

Which of the following best describes an infinite limit?

Accepted Answer

A limit where the function increases or decreases without bound as $$x$$ approaches a certain value.

Question 2

Evaluate $$\lim\limits_{x 	o 0^+} \frac{1}{x}.$$

Accepted Answer

$$\infty$$

Question 3

Which of the following statements is TRUE about vertical asymptotes?

Accepted Answer

A function $$f(x)$$ has a vertical asymptote at $$x=a if$$ $$\lim\limits_{x 	o a^+} f(x) = \infty or$$ $$\lim\limits_{x 	o a^-} f(x) = -\infty.$$

Question 4

Find the vertical asymptotes of $$f(x) = \frac{x+2}{x^2$-4}.$$

Accepted Answer

$$x = 2$$ and $$x = -2$$

Question 5

Evaluate $$\lim\limits_{x 	o 0^-} \frac{1}{x^2$}.$$

Accepted Answer

$$\infty$$

Question 6

Which of the following is the correct definition of a limit at infinity?

Accepted Answer

A limit where $$x$$ approaches infinity and $$f(x)$$ approaches a finite value.

Question 7

Evaluate $$\lim\limits_{x 	o \infty} \frac{3x^2$ + 2}{x^2$ - 5}.$$

Accepted Answer

$$3$$

Question 8

Which of the following functions has a horizontal asymptote at $$y=0$$?

Accepted Answer

$$f(x) = \frac{1}{x}$$

Question 9

For the function $$f(x) = \frac{5x^3$ + 2x}{3x^3$ - x + 4}$$, what is $$\lim\limits_{x 	o \infty} f(x)$$?

Accepted Answer

$$\frac{5}{3}$$

Question 10

A rational function $$f(x) = \frac{p(x)}{q(x)}$$ has a horizontal asymptote at $$y=0 if$$:

Accepted Answer

The degree of $$p(x) is $$less than the degree of $$q(x).$$

Question 11

Evaluate $$\lim\limits_{x 	o -\infty} \frac{2x^2$ - x}{x^2$ + 3}.$$

Accepted Answer

$$2$$

Question 12

Which of the following best describes the behavior of $$f(x) = \frac{1}{x-3} as$$ $$x$$ approaches $$3$$ from the right?

Accepted Answer

$$f(x)$$ approaches $$\infty$$

Question 13

If $$\lim\limits_{x 	o a^-} f(x) = -\infty$$ and $$\lim\limits_{x 	o a^+} f(x) = \infty$$, what can be said about $$x=a$$?

Accepted Answer

$$x=a is a $$vertical asymptote.

Question 14

Evaluate $$\lim\limits_{x 	o \infty} \frac{4x^2$ + 1}{2x^3$ + 5}.$$

Accepted Answer

$$0$$

Question 15

Which of the following is NOT a valid method for evaluating limits at infinity for rational functions?

Accepted Answer

Substituting infinity directly into the function.

Question 16

A function $$f(x)$$ has a horizontal asymptote at $$y=L if$$:

Accepted Answer

$$\lim\limits_{x 	o \infty} f(x) = L$$

Infinite Limits and Limits at Infinity: Study Notes for Calculus

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