Q: Which of the following is true about the function $$f(x) = x + 1$$ and $$g(x) = \frac{x^2$ - 1}{x - 1}$$ for $$x \neq 1$$?

They have the same limit as $$x \to 1.$$

Q: Which of the following is a correct application of the Power Rule for limits?

$$\lim_{x \to c} [f(x)]^n = [\lim_{x \to c} f(x)]^n$$

Question 1

Suppose the position of a falling object is given by $$s(t) = 16t^2$. What is the average speed of the object over the interval $$[2, 2+h]$$?

Accepted Answer

$$\frac{16(2+h)^2$ - 16(2)^2$}{h}$$

Question 2

Given $$f(x) = x^2$, what is the slope of the secant line between x = 2$ and x = 2 + h$?

Accepted Answer

$$\frac{(2+h)^2$ - 4}{h}$$

Question 3

Which of the following best describes the tangent to a curve at a point $$P$$?

Accepted Answer

It is the limit of the secant lines through $$P as $$the second point approaches $$P.$$

Question 4

If $$f(x) = \frac{x^2$ - 1}{x - 1}$$ for $$x 
eq 1$$, what is $$\lim_{x 	o 1} f(x)$$?

Accepted Answer

$$2$$

Question 5

Which of the following is NOT a requirement for $$\lim_{x 	o c} f(x) = L$$ according to the precise (epsilon-delta) definition?

Accepted Answer

There exists $$\delta \u003e 0$$ such that for every $$\epsilon \u003e 0$$, $$0 \u003c |x - c| \u003c \delta$$ implies $$|f(x) - L| \u003c \epsilon.$$

Question 6

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

Accepted Answer

The Sandwich Theorem

Question 7

Given $$\lim_{x 	o c} f(x) = L$$ and $$\lim_{x 	o c} g(x) = M$$, what is $$\lim_{x 	o c} [f(x) + g(x)]$$?

Accepted Answer

$$L + M$$

Question 8

If $$P(x)$$ and $$Q(x)$$ are polynomials and $$Q(c) 
eq 0$$, what is $$\lim_{x 	o c} \frac{P(x)}{Q(x)}$$?

Accepted Answer

$$\frac{P(c)}{Q(c)}$$

Question 9

A fruit fly population grows from 150 to 340 in 22 days. What is the average rate of change in population during this period?

Accepted Answer

$$\frac{340 - 150}{22} = 8.6$$ flies/day

Question 10

Which of the following functions does NOT have a limit as $$x$$ approaches $$0$$?

Accepted Answer

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

Question 11

If $$f(x) = \sqrt{x^2$ + 100} - 10$$ divided by $$x^2$$, what does $$f(x)$$ approach as $$x$$ approaches $$0$$?

Accepted Answer

$$0.05$$

Question 12

Which of the following is true about the function $$f(x) = x + 1$$ and $$g(x) = \frac{x^2$ - 1}{x - 1}$$ for $$x 
eq 1$$?

Accepted Answer

They have the same limit as $$x 	o 1.$$

Question 13

If $$f(x) is $$continuous at $$x = c$$, which of the following must be true?

Accepted Answer

$$\lim_{x 	o c} f(x) = f(c)$$

Question 14

Which of the following is a correct application of the Power Rule for limits?

Accepted Answer

$$\lim_{x 	o c} [f(x)]^n = [\lim_{x 	o c} f(x)]^n$$

Question 15

If $$f(x) is $$sandwiched between $$g(x)$$ and $$h(x)$$, and $$\lim_{x 	o c} g(x) = \lim_{x 	o c} h(x) = L$$, what can you conclude about $$\lim_{x 	o c} f(x)$$?

Accepted Answer

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

Question 16

Which of the following is the correct definition of the average rate of change of $$y = f(x)$$ over $$[x_1$, x_2$]$$?

Accepted Answer

$$\frac{f(x_2$) - f(x_1$)}{x_2$ - x_1$}$$

Limits and Continuity: Chapter 2 Study Guide

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