Question 1

Let $$f(x) = \dfrac{2x^3$ + 5x^2$ - 7}{x^3$ - 4x + 1}. $$What is $$\displaystyle \lim_{x 	o \infty} f(x)$$?

Accepted Answer

$$2$$

Question 2

Which of the following statements is TRUE about the horizontal asymptote of $$f(x) = \dfrac{3x^2$ + 1}{5x^2$ - 2x + 4}$$?

Accepted Answer

$$y = \dfrac{3}{5} is $$the horizontal asymptote.

Question 3

Evaluate $$\displaystyle \lim_{x 	o \infty} \dfrac{4x^2$ + x + 6}{3x^2$ + 7}.$$

Accepted Answer

$$\dfrac{4}{3}$$

Question 4

Suppose $$f(x) = \dfrac{5x^4$ - 2x^2$ + 1}{2x^3$ + 7}. $$What happens to $$f(x) as$$ $$x 	o \infty$$?

Accepted Answer

$$f(x) 	o \infty$$

Question 5

Which of the following best describes the horizontal asymptote(s) of $$f(x) = \dfrac{2x^2$ + 1}{x^3$ + 4}$$?

Accepted Answer

$$y = 0$$

Question 6

A rational function $$f(x) = \dfrac{ax^n + \ldots}{bx^m + \ldots}$$ has $$n = m + 1. $$What type of asymptote does $$f(x)$$ have?

Accepted Answer

A slant (oblique) asymptote

Question 7

Evaluate $$\displaystyle \lim_{x 	o -\infty} \left( -5 - \dfrac{2}{x} ight).$$

Accepted Answer

$$-5$$

Question 8

Which of the following is the correct definition of $$\displaystyle \lim_{x 	o \infty} f(x) = L$$?

Accepted Answer

As $$x$$ increases without bound, $$f(x)$$ approaches the real number $$L.$$

Question 9

Given $$f(x) = \dfrac{2x^2$ + 5}{3x^2$ + 1}$$, what is the end behavior of $$f(x) as$$ $$x 	o -\infty$$?

Accepted Answer

$$f(x) 	o \dfrac{2}{3}$$

Question 10

A chemical reaction's rate is modeled by $$R(t) = \dfrac{10t}{t^2$ + 1}. $$What is $$\displaystyle \lim_{t 	o \infty} R(t)$$ and what does it mean physically?

Accepted Answer

$$0$$; the reaction rate approaches zero as time increases.

Question 11

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

Accepted Answer

$$f(x) = \dfrac{3x + 2}{x^2$ + 1}$$

Question 12

If $$f(x) = \dfrac{4x^2$ + 2x + 1}{2x^2$ - 3x + 5}$$, what is the horizontal asymptote?

Accepted Answer

$$y = 2$$

Question 13

A population model is given by $$P(t) = \dfrac{1000t}{t^2$ + 100}. $$What is the long-term behavior of $$P(t) as$$ $$t 	o \infty$$?

Accepted Answer

$$P(t) 	o 0$$

Question 14

Evaluate $$\displaystyle \lim_{x 	o \infty} \dfrac{3x^3$ - 2x + 1}{x^2$ + 4}.$$

Accepted Answer

$$\infty$$

Question 15

Which of the following statements is TRUE about the function $$f(x) = \dfrac{2x^2$ + 1}{x^2$ + 4}$$?

Accepted Answer

It has a horizontal asymptote at $$y = 2.$$

Question 16

For $$f(x) = \dfrac{5x^2$ + 1}{10x^4$ + x^2$ + 1}$$, what is $$\displaystyle \lim_{x 	o \infty} f(x)$$?

Accepted Answer

$$0$$

Question 17

A rational function $$f(x) = \dfrac{ax^n + \ldots}{bx^m + \ldots}$$ has $$n \u003c m. $$What is the horizontal asymptote?

Accepted Answer

$$y = 0$$

Question 18

If $$f(x) = \dfrac{2x^3$ + 1}{x^2$ + 4}$$, what is the end behavior as $$x 	o -\infty$$?

Accepted Answer

$$f(x) 	o -\infty$$

Question 19

Which of the following is the correct way to find the limit at infinity of a rational function?

Accepted Answer

Divide every term by the highest power of $$x in $$the denominator.

Question 20

A medication's concentration in the bloodstream is modeled by $$C(t) = \dfrac{50}{t + 5}. $$What is $$\displaystyle \lim_{t 	o \infty} C(t)$$ and what does it mean?

Accepted Answer

$$0$$; the concentration approaches zero as time increases.

Limits at Infinity and Asymptotes: Study Notes

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