Q: Which of the following statements about vertical asymptotes is correct?

A vertical asymptote occurs where $$\lim\limits_{x \to a} f(x) = \pm \infty$$

Q: State the Squeeze Theorem.

If $$f(x) \leq g(x) \leq h(x)$$ and $$\lim\limits_{x \to a} f(x) = \lim\limits_{x \to a} h(x) = L$$, then $$\lim\limits_{x \to a} g(x) = L$$

Q: Which of the following is the correct limit definition of the derivative of $$f at$$ $$x = a$$?

$$\lim\limits_{h \to 0} \frac{f(a+h) - f(a)}{h}$$

Q: Which of the following is a correct statement about the end behavior of $$e^x as$$ $$x \to -\infty$$?

$$\lim\limits_{x \to -\infty} e^x = 0$$

Q: Which of the following limits is used to determine if $$y = L is a $$horizontal asymptote of $$f(x)$$?

$$\lim\limits_{x \to \infty} f(x) = L$$

Question 1

A rock is launched vertically upward from the ground with a speed of 96 ft/s. Its position after $$t$$ seconds is given by $$s(t) = -16t^2$ + 96t. $$What is the average velocity of the rock over the interval $$[2, 4]$$ seconds?

Accepted Answer

$$-32 ft/s$$

Question 2

Which of the following is an indeterminate form when evaluating limits?

Accepted Answer

$$\frac{0}{0}$$

Question 3

Evaluate the limit: $$\lim\limits_{x 	o 2} \frac{x^2$ - 6x + 8}{x^2$ - 4}.$$

Accepted Answer

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

Question 4

Which of the following statements about vertical asymptotes is correct?

Accepted Answer

A vertical asymptote occurs where $$\lim\limits_{x 	o a} f(x) = \pm \infty$$

Question 5

Given $$f(x) = \frac{3x + 2}{x^2$ + 1}$$, what is the horizontal asymptote of $$f(x)$$?

Accepted Answer

$$y = 0$$

Question 6

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

Accepted Answer

$$f'(a)$$ exists

Question 7

Which of the following best describes a removable discontinuity?

Accepted Answer

A discontinuity that can be eliminated by redefining $$f(a) so $$that $$f(a) = \lim\limits_{x 	o a} f(x)$$

Question 8

State the Squeeze Theorem.

Accepted Answer

If $$f(x) \leq g(x) \leq h(x)$$ and $$\lim\limits_{x 	o a} f(x) = \lim\limits_{x 	o a} h(x) = L$$, then $$\lim\limits_{x 	o a} g(x) = L$$

Question 9

Which of the following is the correct limit definition of the derivative of $$f at$$ $$x = a$$?

Accepted Answer

$$\lim\limits_{h 	o 0} \frac{f(a+h) - f(a)}{h}$$

Question 10

Suppose $$f(x) is $$continuous on $$[1, 5]$$ and $$f(1) = -2$$, $$f(5) = 6. $$Which theorem guarantees that $$f(x) = 0$$ for some $$x in$$ $$(1, 5)$$?

Accepted Answer

Intermediate Value Theorem

Question 11

Which of the following is a correct statement about the end behavior of $$e^x as$$ $$x 	o -\infty$$?

Accepted Answer

$$\lim\limits_{x 	o -\infty} e^x = 0$$

Question 12

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

Accepted Answer

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

Question 13

Which of the following is true about the function $$f(x) = \frac{2x + 1}{x - 3}$$?

Accepted Answer

It has a vertical asymptote at $$x = 3$$

Question 14

Which of the following is the correct equation for the tangent line to $$f(x) at$$ $$x = a$$ with slope $$m$$?

Accepted Answer

$$y = f(a) + m(x - a)$$

Question 15

If $$\lim\limits_{x 	o a^-} f(x) 
eq \lim\limits_{x 	o a^+} f(x)$$, what type of discontinuity does $$f$$ have at $$x = a$$?

Accepted Answer

Jump discontinuity

Question 16

Which of the following limits is used to determine if $$y = L is a $$horizontal asymptote of $$f(x)$$?

Accepted Answer

$$\lim\limits_{x 	o \infty} f(x) = L$$

Question 17

Given $$f(x) = x^2$ - 9$$, which of the following is the correct factorization?

Accepted Answer

$$(x - 3)(x + 3)$$

Question 18

If $$f(x) is $$not differentiable at $$x = a$$, which of the following could be true?

Accepted Answer

The graph of $$f$$ has a sharp corner at $$x = a$$

Calculus I: Limits, Continuity, and Introduction to the Derivative – Study Notes

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