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

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

Question 1

Suppose $$f(x) = x^2$. What is the slope of the secant line between the points (2$$, 4)$$ and (4$$, 16)$$?

Accepted Answer

$$m = 6$$

Question 2

Given the position function $$s(t)$$ for an object, the average velocity over $$[t_0$, t_1$] is $$given by which formula?

Accepted Answer

$$v_{av} = \frac{s(t_1$) - s(t_0$)}{t_1$ - t_0$}$$

Question 3

What is the instantaneous velocity at $$t = 1$$ for an object whose position function $$s(t)$$ satisfies $$s(1) = 64$$, $$s(1.0001) = 64.00479984$$, $$s(1.001) = 64.047984$$, and $$s(1.01) = 64.4784$$?

Accepted Answer

$$48$$

Question 4

Which of the following best describes the limit $$\lim_{x 	o a} f(x) = L$$?

Accepted Answer

As $$x$$ approaches $$a$$, $$f(x)$$ approaches $$L.$$

Question 5

Evaluate $$\lim_{x 	o 2} (3x - 2). $$Use the limit laws.

Accepted Answer

$$4$$

Question 6

Given $$\lim_{x 	o 2} f(x) = 3$$, $$\lim_{x 	o 2} g(x) = 2$$, and $$\lim_{x 	o 2} h(x) = 5$$, what is $$\lim_{x 	o 2} (4f(x) + 3g(x))$$?

Accepted Answer

$$18$$

Question 7

For the piecewise function $$f(x) = \begin{cases} x^2$ + 3 & x \leq 1 \ 2x + 3 & x \u003e 1 \end{cases}$$, what is $$\lim_{x 	o 1^-} f(x)$$?

Accepted Answer

$$4$$

Question 8

Use the Squeeze Theorem to evaluate $$\lim_{x 	o 0} x^2$ \cos x.$$

Accepted Answer

$$0$$

Question 9

Evaluate $$\lim_{x 	o 0} \frac{\sin 2x}{x}.$$

Accepted Answer

$$2$$

Question 10

Which of the following statements about vertical asymptotes is correct?

Accepted Answer

A vertical asymptote occurs where the denominator of a rational function is zero and the numerator is not zero.

Question 11

Evaluate $$\lim_{x 	o \infty} \frac{2x^2$ + 3}{4x^2$ - 1}.$$

Accepted Answer

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

Question 12

Which of the following is NOT a requirement for a function $$f to be $$continuous at $$x = a$$?

Accepted Answer

$$f(a) is $$differentiable.

Question 13

For which value(s) of $$x is $$the function $$f(x) = \frac{x+1}{x^2$ - 1}$$ NOT continuous?

Accepted Answer

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

Question 14

Which theorem guarantees that if $$f is $$continuous on $$[a, b]$$ and $$L is $$between $$f(a)$$ and $$f(b)$$, then there is some $$c in$$ $$(a, b)$$ with $$f(c) = L$$?

Accepted Answer

Intermediate Value Theorem

Question 15

What is $$\lim_{x 	o -\infty} e^x$$?

Accepted Answer

$$0$$

Question 16

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

Accepted Answer

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

Question 17

If $$\lim_{x 	o 0^+} \frac{1}{x} = \infty$$, what does this mean about the behavior of $$f(x) = \frac{1}{x} as$$ $$x$$ approaches $$0$$ from the right?

Accepted Answer

$$f(x)$$ increases without bound.

Question 18

Which of the following is true about the composition of continuous functions?

Accepted Answer

The composition of two continuous functions is continuous.

Question 19

Suppose $$f(x) is $$continuous on $$[0, 2]$$ and $$f(0) = 1$$, $$f(2) = 5. $$Which value must $$f(x)$$ take for some $$x in$$ $$(0, 2) by $$the Intermediate Value Theorem?

Accepted Answer

$$3$$

Calculus Study Guide: Limits and Continuity (Chapter 2)

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