Q: Determine the end behavior of $$f(x) = 1 - e$$^{-2x}$$ as x$$ \to \infty$$ and x$$ \to -\infty$$.

As $$x \to \infty$$, $$f(x) \to 1$$; as $$x \to -\infty$$, $$f(x) \to -\infty.$$

Question 1

A baseball is thrown upward and its height at time $$t$$ seconds is given by $$s(t) = -16t^2$ + 60t + 6. $$What is the average velocity of the baseball over the interval $$[1, 1.5]$$ seconds?

Accepted Answer

$$20 ft/s$$

Question 2

Given the function $$f(x) is $$defined and continuous everywhere except at $$x=2$$ and $$x=3 on $$the interval $$[-1,4]$$, which of the following statements is true?

Accepted Answer

$$f(x) is $$discontinuous at $$x=2$$ and $$x=3$$

Question 3

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

Accepted Answer

$$2$$

Question 4

Estimate $$\displaystyle \lim_{x 	o \frac{\pi}{4}} \frac{\cos 2x}{\cos x - \sin x}$$ using analytical methods.

Accepted Answer

$$\sqrt{2}$$

Question 5

Calculate $$\displaystyle \lim_{h 	o 0} \frac{\sqrt{5x + h} - \sqrt{5x}}{h}$$ for $$x \u003e 0.$$

Accepted Answer

$$\frac{1}{2\sqrt{5x}}$$

Question 6

Find $$\displaystyle \lim_{p 	o 1} \frac{p^5$ - 1}{p - 1}.$$

Accepted Answer

$$5$$

Question 7

Determine the value of $$\displaystyle \lim_{x 	o 3} \frac{x^4$ - 81}{x - 3}.$$

Accepted Answer

$$108$$

Question 8

Evaluate $$\displaystyle \lim_{x 	o 3^-} \frac{x - 4}{x^2$ - 3x}.$$

Accepted Answer

$$\infty$$

Question 9

What is $$\displaystyle \lim_{x 	o \infty} e$$^{-2x}$$ + \frac{2}{x}$$?

Accepted Answer

$$0$$

Question 10

Find $$\displaystyle \lim_{x 	o \infty} 3 	an$$^{-1}$$ x + 2.$$

Accepted Answer

$$3\frac{\pi}{2} + 2$$

Question 11

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

Accepted Answer

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

Question 12

Which of the following is a vertical asymptote of $$f(x) = \frac{x^2$ - 5x + 6}{x^2$ - 2x}$$?

Accepted Answer

$$x = 0$$

Question 13

Does the function $$h(x) = \begin{cases} -2x + 14 & x \leq 5 \ \sqrt{x^2$ - 9} & x \u003e 5 \end{cases}$$ have continuity at $$x=5$$?

Accepted Answer

Yes, because $$h(5)$$ and the left and right limits at $$x=5$$ are all equal.

Question 14

For $$f(x) = e$$^{\sqrt{x}$$ - 2}$$, on which interval is $$f(x)$$ continuous?

Accepted Answer

$$[2, \infty)$$

Question 15

Is $$h(x) = \sqrt{x^2$ - 9}$$ left-continuous at $$x=3$$?

Accepted Answer

No, because the domain is $$(-\infty, -3] \cup [3, \infty)$$, so left-continuity at $$x=3 is $$not possible.

Question 16

Use the Intermediate Value Theorem to determine if $$g(x) = x^5$ + 7x + 5$$ has a root in $$(-1, 0).$$

Accepted Answer

Yes, because $$g(-1)  0.$$

Question 17

Find all asymptotes of $$f(x) = \frac{x^2$ + x + 3}{|x|}.$$

Accepted Answer

Vertical asymptote at $$x=0$$; slant asymptote $$y=x+1 as$$ $$x 	o \infty$$, $$y=-x-1 as$$ $$x 	o -\infty.$$

Question 18

Determine the end behavior of $$f(x) = 1 - e$$^{-2x}$$ as x$$ 	o \infty$$ and x$$ 	o -\infty$$.

Accepted Answer

As $$x 	o \infty$$, $$f(x) 	o 1$$; as $$x 	o -\infty$$, $$f(x) 	o -\infty.$$

Question 19

A medication's concentration in the bloodstream is modeled by $$C(t) = e$$^{-2t}$$ + \frac{2}{t}$$, where $$t is in $$hours. What happens to the concentration as $$t 	o \infty$$?

Accepted Answer

It approaches $$0.$$

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