Q: Suppose $$f(x)$$ has a vertical asymptote at $$x = 3. $$Which of the following statements is true?

$$\lim\limits_{x \to 3} f(x)$$ does not exist.

Question 1

Which of the following is the correct graph of $$y = \sin(x)$$ for $$x in$$ $$[0, 2\pi]$$?

Accepted Answer

A wave starting at $$(0, 0)$$, reaching a maximum at $$\left(\frac{\pi}{2}, 1\right)$$, a minimum at $$\left(\frac{3\pi}{2}, -1\right)$$, and ending at $$(2\pi, 0).$$

Question 2

Compute the exact value of $$\cos\left(\frac{5\pi}{6}\right).$$

Accepted Answer

$$-\frac{\sqrt{3}}{2}$$

Question 3

Simplify the expression $$	an(x) \cdot \cot(x).$$

Accepted Answer

$$1$$

Question 4

A car travels along a straight road. Its position at time $$t (in $$hours) is given by $$s(t) = 4t^2$ + 2t. $$What is the average velocity between $$t = 1$$ and $$t = 3$$?

Accepted Answer

$$18$$

Question 5

Given the function $$f(x) = \frac{1}{x-2}$$, which of the following statements is true?

Accepted Answer

There is a vertical asymptote at $$x = 2.$$

Question 6

Evaluate $$\lim\limits_{x 	o 0} \frac{\sin(x)}{x}.$$

Accepted Answer

$$1$$

Question 7

Which of the following is a correct application of the Squeeze Theorem?

Accepted Answer

If $$-x^2$$ \leq f(x) \leq $$x^2$$ for all $$x$$ near $$0$$, then $$\lim\limits_{x 	o 0} f(x) = 0.$$

Question 8

Which of the following expressions is equivalent to $$1 - \sin^2$(x)$$?

Accepted Answer

$$\cos^2$(x)$$

Question 9

Given the piecewise function $$f(x) = \begin{cases} x^2$ & x \u003c 1 \ 2x-1 & x \geq 1 \end{cases}$$, find $$\lim\limits_{x 	o 1^-} f(x).$$

Accepted Answer

$$1$$

Question 10

Suppose $$f(x)$$ has a vertical asymptote at $$x = 3. $$Which of the following statements is true?

Accepted Answer

$$\lim\limits_{x 	o 3} f(x)$$ does not exist.

Question 11

If $$	an(	heta) = -1$$ and $$	heta is in $$the second quadrant, what is $$	heta$$?

Accepted Answer

$$\frac{3\pi}{4}$$

Question 12

Estimate the slope of the tangent line to the curve $$y = x^2$ at x = 2$.

Accepted Answer

$$4$$

Question 13

Which of the following is the reciprocal identity for $$\sec(x)$$?

Accepted Answer

$$\sec(x) = \frac{1}{\cos(x)}$$

Question 14

Evaluate $$\lim\limits_{x 	o 2} \frac{x^2$ - 4}{x - 2}.$$

Accepted Answer

$$4$$

Question 15

Which of the following best describes the behavior of $$f(x) = \sin\left(\frac{1}{x}ight) as$$ $$x 	o 0$$?

Accepted Answer

The function oscillates infinitely and the limit does not exist.

Question 16

If $$f(x) is $$continuous everywhere except at $$x = 1$$, where $$f(x)$$ has a vertical asymptote, which of the following could be the equation for $$f(x)$$?

Accepted Answer

$$f(x) = \frac{2}{x-1}$$

Question 17

A medication is administered and its concentration in the bloodstream is modeled by $$C(t) = 10e$$^{-0.5t}$$, where t$ is in hours. What is the average rate of change of C$ from t = 0$ to t = 2$?

Accepted Answer

$$\frac{10e$$^{-1}$$ - 10}{2}$$

Question 18

Which of the following is NOT a Pythagorean identity?

Accepted Answer

$$\sin(x) + \cos(x) = 1$$

Study Guide: Trigonometry and Limits (Calculus I, Sections 2.1–2.3)

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