Q: Given the graph of a rational function with a vertical asymptote at $$x=2$$ and a horizontal asymptote at $$y=1$$, which of the following limit statements is correct for the left end behavior?

$$\lim_{x \to -\infty} f(x) = 1$$

Q: For the function $$h(x) = \frac{2x^2$ - 2x + 1}{3x^2$ + 5x + 7}$$, what is the right end behavior as $$x \to \infty$$?

$$\lim_{x \to \infty} h(x) = \frac{2}{3}$$

Q: For the rational function $$r(x) = \frac{-2x^4$ + 3x^2$ + x - 1}{5x^2$ + 2x + 3}$$, what is the end behavior as $$x \to -\infty$$?

$$\lim_{x \to -\infty} r(x) = -\infty$$

Q: For the function $$f(x) = \frac{-2(x+3)(x+1)}{(x-1)^2$}$$, what is the behavior as $$x$$ approaches 1 from the right?

$$\lim_{x \to 1^+} f(x) = -\infty$$

Q: For $$k(x) = \frac{2x(x-3)}{(x+2)^2$(x-1)}$$, what is the right end behavior as $$x \to \infty$$?

$$\lim_{x \to \infty} k(x) = 0$$

Question 1

Given the graph of a rational function with a vertical asymptote at $$x=2$$ and a horizontal asymptote at $$y=1$$, which of the following limit statements is correct for the left end behavior?

Accepted Answer

$$\lim_{x 	o -\infty} f(x) = 1$$

Question 2

For the function $$h(x) = \frac{2x^2$ - 2x + 1}{3x^2$ + 5x + 7}$$, what is the right end behavior as $$x 	o \infty$$?

Accepted Answer

$$\lim_{x 	o \infty} h(x) = \frac{2}{3}$$

Question 3

A rational function $$k(x) = \frac{2x(x-3)}{(x+2)^2$(x-1)}$$ has which of the following vertical asymptotes?

Accepted Answer

$$x = -2$$ and $$x = 1$$

Question 4

Given the graph of $$f(x)$$ with a vertical asymptote at $$x=2$$ and a horizontal asymptote at $$y=1$$, what is $$\lim_{x 	o 2^-} f(x)$$?

Accepted Answer

$$\infty$$

Question 5

For the rational function $$r(x) = \frac{-2x^4$ + 3x^2$ + x - 1}{5x^2$ + 2x + 3}$$, what is the end behavior as $$x 	o -\infty$$?

Accepted Answer

$$\lim_{x 	o -\infty} r(x) = -\infty$$

Question 6

Which of the following describes the output values as the input values of a rational function increase without bound?

Accepted Answer

$$\lim_{x 	o \infty} f(x)$$

Question 7

Given $$f(x) = \frac{(x-1)(x+5)}{(x-1)(x+2)}$$, what is $$\lim_{x 	o 1^-} f(x)$$?

Accepted Answer

$$6/3 = 2$$

Question 8

For $$y = \frac{(x+3)(x-2)}{(x+3)^2$(x-2)}$$, what type of discontinuity occurs at $$x = -3$$?

Accepted Answer

A hole

Question 9

Solve the inequality $$\frac{x-3}{x+2} \leq 0$$ and express your answer in interval notation.

Accepted Answer

$$(-2, 3]$$

Question 10

For $$r(x) = \frac{x^3$ - x^2$}{x^2$ + 2x + 1}$$, which value of $$x$$ gives a vertical asymptote?

Accepted Answer

$$x = -1$$

Question 11

Given $$m(x) = \frac{3(x-1)^2$(x+5)}{(2x+3)^2$}$$, what is the horizontal asymptote as $$x 	o \infty$$?

Accepted Answer

$$y = 0$$

Question 12

Which of the following is the correct left-hand limit statement for $$f(x) = \frac{(x-2)(x-4)}{(x-1)(x+2)} as$$ $$x$$ approaches 1?

Accepted Answer

$$\lim_{x 	o 1^-} f(x)$$

Question 13

For the function $$f(x) = \frac{-2(x+3)(x+1)}{(x-1)^2$}$$, what is the behavior as $$x$$ approaches 1 from the right?

Accepted Answer

$$\lim_{x 	o 1^+} f(x) = -\infty$$

Question 14

Solve the inequality $$\frac{x^2$ - x - 12}{x^2$} \geq 0.$$

Accepted Answer

$$(-\infty, -3] \cup [4, 0) \cup (0, \infty)$$

Question 15

Which of the following describes the output values as the input values of a rational function decrease without bound?

Accepted Answer

$$\lim_{x 	o -\infty} f(x)$$

Question 16

For $$k(x) = \frac{(x+7)(x+2)^3$}{(x+1)(x+2)^2$}$$, what type of discontinuity occurs at $$x = -2$$?

Accepted Answer

A vertical asymptote

Question 17

Solve the inequality $$\frac{(x-1)^2$(x+2)}{x+1} \u003e 0.$$

Accepted Answer

$$(-\infty, -2) \cup (-1, 1) \cup (1, \infty)$$

Question 18

For $$k(x) = \frac{2x(x-3)}{(x+2)^2$(x-1)}$$, what is the right end behavior as $$x 	o \infty$$?

Accepted Answer

$$\lim_{x 	o \infty} k(x) = 0$$

Question 19

Solve the inequality $$\frac{-2x(x-3)^2$}{(x+2)(x-4)^2$} \u003c 0.$$

Accepted Answer

$$(-\infty, 0) \cup (3, 4)$$

Precalculus Study Guide: Rational Functions, Limits, and End Behavior

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