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

$$h(a) = 0$$ and $$g(a) \neq 0$$

Question 1

Which of the following best describes an infinite limit?

Accepted Answer

As $$x$$ approaches a certain value, $$f(x)$$ increases or decreases without bound.

Question 2

Given $$\lim\limits_{x 	o 2^+} \frac{1}{x-2}$$, what is the value of the limit?

Accepted Answer

$$+\infty$$

Question 3

Which of the following statements is TRUE about vertical asymptotes?

Accepted Answer

A vertical asymptote occurs where the denominator of a rational function is zero and the numerator is not zero.

Question 4

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

Accepted Answer

$$-\infty$$

Question 5

If $$\lim\limits_{x 	o a} f(x) = \infty$$, what does this imply about the graph of $$f(x) at$$ $$x=a$$?

Accepted Answer

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

Question 6

Which of the following limits represents a limit at infinity?

Accepted Answer

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

Question 7

Suppose $$f(x) = \frac{x^2$}{x-1}. $$What is the vertical asymptote of $$f(x)$$?

Accepted Answer

$$x=1$$

Question 8

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

Accepted Answer

$$+\infty$$

Question 9

A rational function $$f(x) = \frac{g(x)}{h(x)}$$ has a vertical asymptote at $$x=a if$$:

Accepted Answer

$$h(a) = 0$$ and $$g(a) 
eq 0$$

Question 10

Which of the following best describes the behavior of $$f(x) = \frac{1}{x^2$} as$$ $$x 	o \infty$$?

Accepted Answer

$$f(x)$$ approaches $$0$$

Question 11

Given $$f(x) = \frac{x^2$-9}{x+3}$$, what is $$\lim\limits_{x 	o -3} f(x)$$?

Accepted Answer

$$0$$

Question 12

Which of the following is NOT a possible result of an infinite limit?

Accepted Answer

The function approaches a finite value as $$x$$ approaches a value.

Question 13

If $$\lim\limits_{x 	o 1^-} \frac{1}{x-1} = -\infty$$ and $$\lim\limits_{x 	o 1^+} \frac{1}{x-1} = +\infty$$, what can you conclude?

Accepted Answer

There is a vertical asymptote at $$x=1.$$

Question 14

Evaluate $$\lim\limits_{x 	o 2} \frac{x^2$-4}{x-2}.$$

Accepted Answer

$$4$$

Question 15

Which of the following best describes the limit $$\lim\limits_{x 	o 0^-} \frac{1}{x}$$?

Accepted Answer

$$-\infty$$

Question 16

A chemical reaction rate is modeled by $$R(t) = \frac{1}{t-5}. $$What happens to the reaction rate as $$t$$ approaches $$5$$ from the right?

Accepted Answer

The reaction rate approaches $$+\infty.$$

Infinite Limits and Limits at Infinity

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