Q: Which of the following statements best describes a horizontal asymptote of a function $$f(x)$$?

A line $$y = L$$ such that $$\lim_{x \to \infty} f(x) = L or$$ $$\lim_{x \to -\infty} f(x) = L.$$

Q: Which of the following limits implies that $$x = c is a $$vertical asymptote of $$f(x)$$?

$$\lim_{x \to c^+} f(x) = \infty$$

Question 1

Which of the following statements best describes a horizontal asymptote of a function $$f(x)$$?

Accepted Answer

A line $$y = L$$ such that $$\lim_{x 	o \infty} f(x) = L or$$ $$\lim_{x 	o -\infty} f(x) = L.$$

Question 2

Evaluate $$\displaystyle \lim_{x 	o \infty} \frac{5x^2$ + 8x + 3}{x^2$}.$$

Accepted Answer

$$5$$

Question 3

A function $$f(x)$$ has $$\displaystyle \lim_{x 	o \infty} f(x) = 2$$ and $$\displaystyle \lim_{x 	o -\infty} f(x) = -3. $$How many horizontal asymptotes does $$f(x)$$ have?

Accepted Answer

Two

Question 4

Find the horizontal asymptotes of $$f(x) = \frac{\sqrt{4x^2$ + 5x + 1}}{x}.$$

Accepted Answer

$$y = 2$$ and $$y = -2$$

Question 5

Which of the following limits implies that $$x = c is a $$vertical asymptote of $$f(x)$$?

Accepted Answer

$$\lim_{x 	o c^+} f(x) = \infty$$

Question 6

Evaluate $$\displaystyle \lim_{x 	o \infty} \frac{3x^3$ + 2x^2$ + 1}{2x^3$ + 5}.$$

Accepted Answer

$$\frac{3}{2}$$

Question 7

Suppose $$f(x) = \frac{1}{x^2$ - 1}. $$Which of the following are vertical asymptotes of $$f(x)$$?

Accepted Answer

$$x = 1$$ and $$x = -1$$

Question 8

Evaluate $$\displaystyle \lim_{x 	o \infty} e$$^{-x}$$.

Accepted Answer

$$0$$

Question 9

Find $$\displaystyle \lim_{x 	o \infty} \arctan(x).$$

Accepted Answer

$$\frac{\pi}{2}$$

Question 10

A rational function $$f(x)$$ has a higher degree in the denominator than in the numerator. What is $$\displaystyle \lim_{x 	o \infty} f(x)$$?

Accepted Answer

$$0$$

Question 11

Evaluate $$\displaystyle \lim_{x 	o \infty} \frac{2e^x + 3}{4e^x + 5}.$$

Accepted Answer

$$\frac{1}{2}$$

Question 12

Which of the following best describes the end behavior of $$y = x$$^{0.000001}$$ as x$$ 	o \infty$$?

Accepted Answer

Increases without bound, but very slowly

Question 13

Suppose $$f(x) = \frac{3x^3$ + 2x^2$ + 1}{2x^3$ + 5}. $$What is the horizontal asymptote of $$f(x)$$?

Accepted Answer

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

Question 14

Evaluate $$\displaystyle \lim_{x 	o \infty} \frac{1}{x^2$ - 1}.$$

Accepted Answer

$$0$$

Question 15

A function $$f(x)$$ has $$\displaystyle \lim_{x 	o \infty} f(x) = 5$$ and $$\displaystyle \lim_{x 	o -\infty} f(x) = 5. $$What can you conclude about the horizontal asymptotes?

Accepted Answer

There is one horizontal asymptote at $$y = 5$$

Question 16

Find $$\displaystyle \lim_{x 	o \infty} \frac{10x^2$ + 10}{x}.$$

Accepted Answer

$$\infty$$

Question 17

Which of the following is true about the number of horizontal asymptotes a function can have?

Accepted Answer

A function can have zero, one, or two horizontal asymptotes, but not three.

Question 18

Evaluate $$\displaystyle \lim_{x 	o \infty} \frac{\sqrt{x^4$ + 5x + 1}}{x^2$}.$$

Accepted Answer

$$1$$

Question 19

Suppose $$f(x) = \frac{2e^x + 3}{4e^x + 5}. $$What is the horizontal asymptote as $$x 	o -\infty$$?

Accepted Answer

$$y = 0$$

Question 20

A patient’s medication concentration is modeled by $$C(t) = \frac{5}{t^2$ + 1}$$, where $$t is $$time in hours. What happens to the concentration as $$t 	o \infty$$?

Accepted Answer

It approaches $$0$$

Limits Involving Infinity and Asymptotes: Study Notes

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