Q: If $$\lim_{x \to a^-} f(x) = 3$$ and $$\lim_{x \to a^+} f(x) = 5$$, what can you conclude about $$\lim_{x \to a} f(x)$$?

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

Q: Which of the following is a correct application of the limit laws for $$\lim_{x \to a} [f(x) + g(x)]$$?

$$\lim_{x \to a} f(x) + \lim_{x \to a} g(x)$$

Q: Suppose $$f(x) is $$continuous at $$x = a. $$Which of the following must be true?

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

Question 1

A rock is thrown upward from the ground with an initial velocity of 96 ft/s. Its height above the ground after $$t$$ seconds is given by $$s(t) = -16t^2$ + 96t. $$What is the average velocity of the rock over the interval $$[1, 2]$$ seconds?

Accepted Answer

$$48\ 	ext{ft/s}$$

Question 2

Given the function $$f(x) = \frac{x^3$ - 8}{4(x-2)}$$, what is $$\lim_{x 	o 2} f(x)$$?

Accepted Answer

$$3$$

Question 3

Which of the following best describes the instantaneous velocity of the rock at $$t = 1$$ second, given $$s(t) = -16t^2$ + 96t$$?

Accepted Answer

It is the limit of the average velocity as the time interval shrinks to zero at $$t = 1.$$

Question 4

Evaluate $$\lim_{x 	o 1} \frac{\sqrt{x} - 1}{x - 1}.$$

Accepted Answer

$$\frac{1}{2}$$

Question 5

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

Accepted Answer

Squeeze Theorem

Question 6

Suppose $$f(x) is $$defined as $$f(x) = \begin{cases} -2x + 4 & x \leq 1 \ \sqrt{x-1} & x \u003e 1 \end{cases}. $$What is $$\lim_{x 	o 1^-} f(x)$$?

Accepted Answer

$$2$$

Question 7

Which of the following statements is true about $$\lim_{x 	o 0} \cos\left(\frac{1}{x}ight)$$?

Accepted Answer

The limit does not exist because the function oscillates between $$-1$$ and $$1 as$$ $$x 	o 0.$$

Question 8

Given $$f(x) = mx + b$$, what is $$\lim_{x 	o a} f(x)$$?

Accepted Answer

$$ma + b$$

Question 9

If $$\lim_{x 	o a^-} f(x) = 3$$ and $$\lim_{x 	o a^+} f(x) = 5$$, what can you conclude about $$\lim_{x 	o a} f(x)$$?

Accepted Answer

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

Question 10

Which of the following is a correct application of the limit laws for $$\lim_{x 	o a} [f(x) + g(x)]$$?

Accepted Answer

$$\lim_{x 	o a} f(x) + \lim_{x 	o a} g(x)$$

Question 11

Suppose $$f(x) is $$continuous at $$x = a. $$Which of the following must be true?

Accepted Answer

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

Question 12

Evaluate $$\lim_{x 	o 0} \frac{\sin x}{x}$$ using the Squeeze Theorem.

Accepted Answer

$$1$$

Question 13

If $$\lim_{x 	o 2} f(x) = 3$$ and $$\lim_{x 	o 2} g(x) = 4$$, what is $$\lim_{x 	o 2} [f(x)g(x)]$$?

Accepted Answer

$$12$$

Question 14

Given $$f(x) = x^2$ and g(x) = x + 1$, find $$\lim_{x 	o 3} \frac{f(x)}{g(x)}$$.

Accepted Answer

$$\frac{9}{4}$$

Question 15

Which of the following is NOT a requirement for the Squeeze Theorem to apply?

Accepted Answer

$$g(x)$$ must be continuous at $$x = a$$

Question 16

If $$\lim_{x 	o a} f(x)$$ exists but $$f(a) is $$undefined, which of the following is true?

Accepted Answer

The limit exists but the function is not defined at $$x = a.$$

Question 17

Evaluate $$\lim_{x 	o 0^+} 2^x.$$

Accepted Answer

$$1$$

Question 18

Suppose $$f(x) is $$defined for all $$x$$ near $$a$$ except possibly at $$a. If$$ $$f(x) is $$arbitrarily close to $$L$$ for all $$x$$ sufficiently close to $$a$$, what can we say?

Accepted Answer

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

Limits: Concepts, Definitions, and Computation Techniques

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