A line $$x = a$$ where $$\lim_{x \to a} f(x) = \pm \infty$$

Question 1

Which of the following best describes an infinite limit as $$x$$ approaches $$a$$?

Accepted Answer

The function $$f(x)$$ increases or decreases without bound as $$x$$ approaches $$a.$$

Question 2

Evaluate $$\lim_{x 	o 2} \frac{1}{(x-2)^2$}.$$

Accepted Answer

$$+\infty$$

Question 3

What is the value of $$\lim_{x 	o 0} \frac{1}{x^2$}$$?

Accepted Answer

$$+\infty$$

Question 4

For which value of $$a$$ does $$\lim_{x 	o a} \frac{1}{(x-a)^n}$$ equal $$+\infty$$ for all even $$n$$?

Accepted Answer

For all $$a \in \mathbb{R}$$

Question 5

What is the result of $$\lim_{x 	o 4} \frac{1}{(x-4)^2$}$$?

Accepted Answer

$$+\infty$$

Question 6

Which of the following statements about $$\lim_{x 	o 0} \frac{1}{x} is $$correct?

Accepted Answer

The limit does not exist because the function approaches $$+\infty$$ from the right and $$-\infty$$ from the left.

Question 7

If $$\lim_{x 	o a} f(x) = +\infty$$, what does this mean about the behavior of $$f(x)$$ near $$x = a$$?

Accepted Answer

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

Question 8

Which of the following is true about $$\lim_{x 	o a} \frac{1}{(x-a)^2$}$$?

Accepted Answer

The limit is $$+\infty$$ regardless of the direction from which $$x$$ approaches $$a.$$

Question 9

Consider the function $$f(x) = \frac{1}{x^2$}. $$What happens to $$f(x) as$$ $$x$$ approaches $$0$$ from either side?

Accepted Answer

$$f(x)$$ approaches $$+\infty$$ from both sides.

Question 10

Evaluate $$\lim_{x 	o 0} \frac{\sin(x)}{x}.$$

Accepted Answer

$$1$$

Question 11

Which of the following best describes a vertical asymptote?

Accepted Answer

A line $$x = a$$ where $$\lim_{x 	o a} f(x) = \pm \infty$$

Question 12

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

Accepted Answer

$$f(x)$$ has a vertical asymptote at $$x = a$$

Question 13

Which of the following is a real-world example of an infinite limit?

Accepted Answer

The pressure in a container as its volume approaches zero.

Question 14

What is the correct interpretation of $$\lim_{x 	o a} f(x) = -\infty$$?

Accepted Answer

As $$x$$ approaches $$a$$, $$f(x)$$ decreases without bound.

Question 15

Which of the following functions has an infinite limit at $$x = 0$$?

Accepted Answer

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

Question 16

If $$\lim_{x 	o a} \frac{1}{(x-a)^n}$$, where $$n is $$odd, what happens as $$x$$ approaches $$a$$ from the left?

Accepted Answer

The limit is $$-\infty$$

Question 17

Which of the following is NOT a property of infinite limits?

Accepted Answer

The function approaches a finite value as $$x$$ approaches $$a.$$

Question 18

What is the behavior of $$\frac{1}{(x-a)^2$} as$$ $$x$$ approaches $$a$$?

Accepted Answer

It approaches $$+\infty$$ from both sides.

Question 19

Which of the following limits is used to define a vertical asymptote?

Accepted Answer

$$\lim_{x 	o a} f(x) = \pm \infty$$

Limites Infinitos (Infinite Limits) – Calculus Study Notes

Study Guide - Practice Questions