Q: Which of the following is a necessary condition for a function $$f to be $$continuous at $$x = a$$?

$$\lim_{x \to a} f(x)$$ exists and equals $$f(a)$$

Q: A rational function $$f(x) = \frac{p(x)}{q(x)}$$ has a vertical asymptote at $$x = a if$$:

$$q(a) = 0$$ and $$p(a) \neq 0$$

Q: Which of the following best describes a horizontal asymptote of a function $$f(x)$$?

A line $$y = L$$ where $$\lim_{x \to \infty} f(x) = L$$

Question 1

Suppose a rock is thrown upward and its height above the ground at time $$t$$ seconds is given by $$s(t) = -16t^2$ + 96t. $$What is the average velocity of the rock between $$t = 1$$ and $$t = 3$$ seconds?

Accepted Answer

$$32 ft/s$$

Question 2

Evaluate $$\lim_{x 	o 2} \frac{x^2$ - 8}{4(x-2)}.$$

Accepted Answer

$$3$$

Question 3

Which of the following statements is TRUE about the limit $$\lim_{x 	o a} f(x) = L$$?

Accepted Answer

As $$x$$ approaches $$a$$, $$f(x)$$ gets arbitrarily close to $$L.$$

Question 4

Given $$f(x) = \frac{1}{x^2$}$$, what is $$\lim_{x 	o 0} f(x)$$?

Accepted Answer

$$\infty$$

Question 5

Which of the following is a necessary condition for a function $$f to be $$continuous at $$x = a$$?

Accepted Answer

$$\lim_{x 	o a} f(x)$$ exists and equals $$f(a)$$

Question 6

Evaluate $$\lim_{x 	o \infty} \frac{3x^2$ + 2}{x^2$ - 5}.$$

Accepted Answer

$$3$$

Question 7

Which theorem allows you to conclude $$\lim_{x 	o a} g(x) = L if$$ $$f(x) \leq g(x) \leq h(x)$$ and $$\lim_{x 	o a} f(x) = \lim_{x 	o a} h(x) = L$$?

Accepted Answer

Squeeze Theorem

Question 8

If $$\lim_{x 	o 1^-} f(x) = 2$$ and $$\lim_{x 	o 1^+} f(x) = 0$$, what can you conclude about $$\lim_{x 	o 1} f(x)$$?

Accepted Answer

The limit does not exist.

Question 9

Which of the following functions is continuous everywhere in its domain?

Accepted Answer

$$\sin(x)$$

Question 10

A rational function $$f(x) = \frac{p(x)}{q(x)}$$ has a vertical asymptote at $$x = a if$$:

Accepted Answer

$$q(a) = 0$$ and $$p(a) 
eq 0$$

Question 11

Evaluate $$\lim_{x 	o 0^+} \cos\left(\frac{1}{x}ight).$$

Accepted Answer

The limit does not exist.

Question 12

Which of the following best describes a horizontal asymptote of a function $$f(x)$$?

Accepted Answer

A line $$y = L$$ where $$\lim_{x 	o \infty} f(x) = L$$

Question 13

If $$f(x) is $$continuous on $$[a, b]$$ and $$L is $$between $$f(a)$$ and $$f(b)$$, which theorem guarantees there is $$c in$$ $$(a, b)$$ such that $$f(c) = L$$?

Accepted Answer

Intermediate Value Theorem

Question 14

Evaluate $$\lim_{x 	o 0} x^2$ \sin\left(\frac{1}{x}ight).$$

Accepted Answer

$$0$$

Question 15

Which of the following is the correct definition of the limit $$\lim_{x 	o a} f(x) = L$$ using $$\epsilon$$ and $$\delta$$?

Accepted Answer

For every $$\epsilon \u003e 0$$, there exists $$\delta \u003e 0$$ such that if $$0 \u003c |x - a| \u003c \delta$$, then $$|f(x) - L| \u003c \epsilon$$

Question 16

Which of the following is NOT a requirement for continuity of $$f at$$ $$x = a$$?

Accepted Answer

$$f(x) is $$differentiable at $$x = a$$

Question 17

Suppose $$f(x) is $$defined as $$f(x) = \frac{x^2$ - 4x + 3}{x^2$ - 1}. $$What are the vertical asymptotes of $$f(x)$$?

Accepted Answer

$$x = 1$$ and $$x = -1$$

Question 18

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

Accepted Answer

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

Limits in Calculus: Concepts, Definitions, and Techniques

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