Q: Which of the following is the correct notation for the limit of $$f(x) as$$ $$x$$ approaches $$a$$ from the left?

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

Q: Which of the following statements is true about the function $$f(x) = \frac{1}{x}$$?

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

Q: Which of the following is NOT a correct interpretation of the arrow notation?

$$x \to -a$$: x$ approaches negative infinity.

Question 1

Which of the following best describes an infinite limit?

Accepted Answer

The function values increase or decrease without bound as $$x$$ approaches a specific value.

Question 2

Given $$f(x) = \frac{1}{x-2}$$, what happens to $$f(x) as$$ $$x$$ approaches $$2$$ from the right?

Accepted Answer

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

Question 3

Which of the following is the correct notation for the limit of $$f(x) as$$ $$x$$ approaches $$a$$ from the left?

Accepted Answer

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

Question 4

For the function $$f(x) = \frac{3}{x-2}$$, what is the vertical asymptote?

Accepted Answer

$$x = 2$$

Question 5

Evaluate $$\lim\limits_{x 	o 1^-} \frac{1}{x-1}.$$

Accepted Answer

$$-\infty$$

Question 6

A rational function $$f(x)$$ has a vertical asymptote at $$x = -2$$ and a hole at $$x = 2. $$Which of the following could be $$f(x)$$?

Accepted Answer

$$f(x) = \frac{x^2$ + 2x - 8}{x^2$ - 4}$$

Question 7

What is the value of $$\lim\limits_{x 	o 1} \frac{x^2$ - 4x + 3}{x^2$ - 1}$$?

Accepted Answer

$$-1$$

Question 8

Which of the following statements is true about the function $$f(x) = \frac{1}{x}$$?

Accepted Answer

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

Question 9

If $$\lim\limits_{x 	o a} f(x) = \infty$$, what does this imply about the graph of $$f(x) at$$ $$x = a$$?

Accepted Answer

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

Question 10

Evaluate $$\lim\limits_{x 	o 3^-} \frac{2-5x}{x-3}.$$

Accepted Answer

$$\infty$$

Question 11

Which of the following best describes a horizontal asymptote?

Accepted Answer

A horizontal line that the graph approaches as $$x$$ approaches $$\pm\infty.$$

Question 12

Given $$g(x) = \frac{x-2}{(x-1)^2$(x-3)}$$, what is $$\lim\limits_{x 	o 3^-} g(x)$$?

Accepted Answer

$$-\infty$$

Question 13

Which of the following is true about $$\lim\limits_{x 	o -1^-} \frac{x-3}{x+1}$$?

Accepted Answer

$$\infty$$

Question 14

A chemical reaction rate is modeled by $$R(t) = \frac{1}{t-5}$$, where $$t is $$time in seconds. What happens to the reaction rate as $$t$$ approaches $$5$$ from the left?

Accepted Answer

The reaction rate approaches $$-\infty.$$

Question 15

Evaluate $$\lim\limits_{x 	o -4^-} \frac{-x^3$ + 5x^2$ - 6x}{-x^3$ - 4x^2$}.$$

Accepted Answer

$$-\infty$$

Question 16

Which of the following is NOT a correct interpretation of the arrow notation?

Accepted Answer

$$x 	o -a$$: x$ approaches negative infinity.

2.4 Infinite Limits and Vertical Asymptotes

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