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

$$\lim\limits_{x \to a^-} f(x) = \infty$$

Q: Suppose $$f(x)$$ has a vertical asymptote at $$x=0. $$Which of the following could describe the behavior of $$f(x)$$ near $$x=0$$?

$$\lim\limits_{x \to 0^-} f(x) = -\infty$$

Question 1

Which of the following best describes the meaning of $$\lim\limits_{x 	o a^-} f(x) = L$$?

Accepted Answer

As $$x$$ approaches $$a$$ from the left, $$f(x)$$ approaches $$L.$$

Question 2

Given $$f(x) = \frac{2(x^2$-2)}{x^2$-5}$$, what are the vertical asymptotes of $$f(x)$$?

Accepted Answer

$$x = 5$$ and $$x = -5$$

Question 3

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

Accepted Answer

$$\lim\limits_{x 	o a^-} f(x) = \infty$$

Question 4

A rational function $$R(x) in $$lowest terms has a vertical asymptote at $$x = r if$$:

Accepted Answer

$$r is a $$real zero of the denominator

Question 5

Which of the following functions has a hole (removable discontinuity) at $$x = 1$$?

Accepted Answer

$$c(x) = \frac{x^2$-1}{x-1}$$

Question 6

If $$\lim\limits_{x 	o 0^+} f(x) = \infty$$, what does this tell you about the graph of $$f(x)$$?

Accepted Answer

There is a vertical asymptote at $$x = 0$$

Question 7

Which of the following rational functions has no vertical asymptotes?

Accepted Answer

$$b(x) = \frac{2}{x^2$+1}$$

Question 8

Given $$H(x) = \frac{3x^2$ - x}{x+1}$$, what is the equation of the slant (oblique) asymptote?

Accepted Answer

$$y = 3x + 3$$

Question 9

Why does $$c(x) = \frac{x^2$-1}{x-1}$$ have a hole at $$x=1$$ and not a vertical asymptote?

Accepted Answer

Because $$x-1$$ cancels with a factor in the numerator, leaving no denominator at $$x=1.$$

Question 10

Which of the following statements is TRUE about the function $$f(x) = \frac{2(x^2$-2)}{x^2$-5}$$?

Accepted Answer

It has vertical asymptotes at $$x=\sqrt{5}$$ and $$x=-\sqrt{5}.$$

Question 11

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

Accepted Answer

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

Question 12

Which of the following best describes an oblique (slant) asymptote?

Accepted Answer

A line $$y = mx + b$$ that the graph of $$R(x)$$ approaches as $$x 	o \infty or$$ $$x 	o -\infty$$ when the degree of the numerator is one more than the degree of the denominator.

Question 13

Which of the following is NOT a necessary condition for a rational function to have a vertical asymptote at $$x = r$$?

Accepted Answer

$$r is a $$real zero of the numerator.

Question 14

Suppose $$f(x)$$ has a vertical asymptote at $$x=0. $$Which of the following could describe the behavior of $$f(x)$$ near $$x=0$$?

Accepted Answer

$$\lim\limits_{x 	o 0^-} f(x) = -\infty$$

Vertical and Oblique Asymptotes of Rational Functions

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