Q: Which of the following statements about vertical asymptotes is correct?

A vertical asymptote occurs at $$x = a if$$ $$\lim_{x \to a} f(x) = \infty or$$ $$-\infty.$$

Q: If $$\lim_{x \to a^-} f(x) = -\infty$$ and $$\lim_{x \to a^+} f(x) = \infty$$, what can you conclude about $$\lim_{x \to a} f(x)$$?

$$\lim_{x \to a} f(x)$$ does not exist (DNE)

Question 1

Which of the following statements about vertical asymptotes is correct?

Accepted Answer

A vertical asymptote occurs at $$x = a if$$ $$\lim_{x 	o a} f(x) = \infty or$$ $$-\infty.$$

Question 2

Evaluate $$\lim_{x 	o 0^+} \frac{1}{x}.$$

Accepted Answer

$$\infty$$

Question 3

Given $$f(x) = \frac{2 - 5x}{x - 3}$$, what is $$\lim_{x 	o 3^+} f(x)$$?

Accepted Answer

$$-\infty$$

Question 4

For $$f(x) = \frac{x^2$ - 3x + 2}{x - 3}$$, what is $$\lim_{x 	o 3^+} f(x)$$?

Accepted Answer

$$\infty$$

Question 5

Suppose $$f(x) = \frac{x - 7}{x(x + 5)}. $$What is $$\lim_{x 	o -5^-} f(x)$$?

Accepted Answer

$$-\infty$$

Question 6

Let $$f(x) = \frac{x^2$ - 8x + 15}{x^2$ - 5x + 6}. $$Which values of $$x$$ are vertical asymptotes?

Accepted Answer

$$x = 3$$ and $$x = 2$$

Question 7

Evaluate $$\lim_{x 	o 2^+} \frac{x^2$ - 8x + 15}{x^2$ - 5x + 6}.$$

Accepted Answer

$$-\infty$$

Question 8

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

Accepted Answer

$$\lim_{x 	o a} f(x)$$ does not exist (DNE)

Question 9

A chemical reaction rate is modeled by $$R(t) = \frac{1}{t - 2}$$, where $$t is $$time in seconds. What happens to $$R(t) as$$ $$t$$ approaches $$2$$ from the right?

Accepted Answer

$$R(t)$$ approaches $$\infty$$

Question 10

Which of the following best describes the behavior of $$f(x) = \frac{1}{x} as$$ $$x$$ approaches $$0$$ from the left?

Accepted Answer

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

Question 11

A medical dosage is modeled by $$D(x) = \frac{2 - 5x}{x - 3}. $$What is the limit of $$D(x) as$$ $$x$$ approaches $$3$$ from the left?

Accepted Answer

$$\infty$$

Question 12

For $$f(x) = \frac{-4}{(x - 5)^2$}$$, what is $$\lim_{x 	o 5^+} f(x)$$?

Accepted Answer

$$-\infty$$

Question 13

Which of the following is true about the function $$f(x) = \frac{x^2$ - 8x + 15}{x^2$ - 5x + 6} at$$ $$x = 5$$?

Accepted Answer

$$f(x) is $$undefined at $$x = 5$$ but does not have a vertical asymptote there

Question 14

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

Accepted Answer

$$-\infty$$

Question 15

A population model is given by $$P(t) = \frac{2t - 15}{t - 3}. $$What is $$\lim_{t 	o 3^+} P(t)$$?

Accepted Answer

$$-\infty$$

Question 16

Which of the following is a correct interpretation of $$\lim_{x 	o a} f(x) = \infty in a $$real-world context?

Accepted Answer

The value of $$f(x)$$ becomes arbitrarily large as $$x$$ approaches $$a.$$

Question 17

If $$\lim_{x 	o 3^-} \frac{2 - 5x}{x - 3} = \infty$$ and $$\lim_{x 	o 3^+} \frac{2 - 5x}{x - 3} = -\infty$$, what does this indicate about the graph at $$x = 3$$?

Accepted Answer

There is a vertical asymptote at $$x = 3.$$

Question 18

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

Accepted Answer

$$f(x)$$ approaches $$\infty$$

Question 19

A function $$f(x)$$ has $$\lim_{x 	o a^-} f(x) = \infty$$ and $$\lim_{x 	o a^+} f(x) = \infty. $$What is $$\lim_{x 	o a} f(x)$$?

Accepted Answer

$$\infty$$

Infinite Limits and Vertical Asymptotes 2.4

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