$$\lim_{x \to 0} \frac{\sin(x)}{x}$$

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 the following limit: $$\lim_{x 	o 2} \frac{1}{(x-2)^2$}$$

Accepted Answer

$$\infty$$

Question 3

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

Accepted Answer

$$\infty$$

Question 4

Which of the following statements is true about $$\lim_{x 	o a} \frac{1}{(x-a)^n}$$ for any positive integer $$n$$?

Accepted Answer

The limit is $$\infty$$ for all $$a \in \mathbb{R}.$$

Question 5

Consider the function $$f(x) = \frac{1}{(x-4)^2$}. $$What happens to $$f(x) as$$ $$x$$ approaches $$4$$?

Accepted Answer

It approaches $$\infty.$$

Question 6

Which of the following is the correct interpretation of $$\lim_{x 	o 0} \frac{\sin(x)}{x}$$?

Accepted Answer

$$1$$

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 an example of a function with an infinite limit at $$x = 0$$?

Accepted Answer

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

Question 9

What is the left-hand limit $$\lim_{x 	o 0^-} \frac{1}{x}$$?

Accepted Answer

$$-\infty$$

Question 10

What is the right-hand limit $$\lim_{x 	o 0^+} \frac{1}{x}$$?

Accepted Answer

$$\infty$$

Question 11

A medical dosage function is modeled by $$f(x) = \frac{1}{(x-5)^2$}. $$What happens to the dosage as $$x$$ approaches $$5$$?

Accepted Answer

The dosage approaches $$\infty.$$

Question 12

If $$\lim_{x 	o a} f(x) = -\infty$$, which of the following is true?

Accepted Answer

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

Question 13

Which of the following limits does NOT result in an infinite limit?

Accepted Answer

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

Question 14

In the context of infinite limits, what does it mean for a function to 'blow up' at a point?

Accepted Answer

The function increases or decreases without bound as it approaches that point.

Question 15

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

Accepted Answer

The intensity of light as you approach the center of a laser beam modeled by $$I(x) = \frac{1}{x^2$}.$$

Infinite Limits and Their Properties

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