Q: Which of the following limits does NOT exist?

$$\displaystyle \lim_{x \to \frac{\pi}{2}^-} \csc x$$

Question 1

Which of the following is the domain of the function $$f(x) = \sqrt{x^2$ - 4}$$?

Accepted Answer

$$(-\infty, -2] \cup [2, \infty)$$

Question 2

Determine whether the function $$f(x) = x^3$ - x is $$even, odd, or neither.

Accepted Answer

Odd

Question 3

Which of the following functions has an inverse, and what is its inverse?

Accepted Answer

$$f(x) = x^2$ for x$$ \geq 0$$; inverse is f$$^{-1}$$(x) = \sqrt{x}$$

Question 4

Simplify $$\sin(\arccos x)$$ for $$x \in [-1, 1].$$

Accepted Answer

$$\sqrt{1 - x^2$}$$

Question 5

Evaluate $$\displaystyle \lim_{x 	o 1} \frac{x^3$ - 1}{x - 1}.$$

Accepted Answer

$$3$$

Question 6

For which value of $$a is $$the function $$f(x) = \begin{cases} a x^2$ & x \geq 2 \ a x + 2 & x \u003c 2 \end{cases}$$ continuous at $$x = 2$$?

Accepted Answer

$$a = 1$$

Question 7

Which of the following functions has a vertical asymptote at $$x = 1$$?

Accepted Answer

$$f(x) = \frac{x^2$ - 4x + 3}{x - 1}$$

Question 8

Using the limit definition, find $$f'(x)$$ for $$f(x) = x^2$.

Accepted Answer

$$2x$$

Question 9

Find the equation of the tangent line to $$f(x) = x^3$ at the point (1$$, 1)$$.

Accepted Answer

$$y = 3x - 2$$

Question 10

Find the second derivative of $$f(x) = x^3$ + 3x^2$.

Accepted Answer

$$f''(x) = 6x + 6$$

Question 11

Evaluate $$\displaystyle \lim_{x 	o 0} \frac{e^x - 1}{x}.$$

Accepted Answer

$$1$$

Question 12

Which of the following best describes the horizontal asymptote of $$f(x) = \frac{3x^2$ - 4x + 1}{x^2$ - 2x + 1}$$?

Accepted Answer

$$y = 3$$

Question 13

Find the derivative of $$f(x) = e^x x^4$ using the product rule.

Accepted Answer

$$e^x x^4$ + 4x^3$ e^x$$

Question 14

If $$f(x) = \ln(x)$$, what is $$f'(x)$$?

Accepted Answer

$$\frac{1}{x}$$

Question 15

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

Accepted Answer

$$3$$

Question 16

Find the derivative of $$f(x) = \frac{e^x}{x^2$}$$ using the quotient rule.

Accepted Answer

$$\frac{e^x(x^2$ - 2x)}{x^4$}$$

Question 17

Which of the following limits does NOT exist?

Accepted Answer

$$\displaystyle \lim_{x 	o \frac{\pi}{2}^-} \csc x$$

Question 18

Suppose $$f(x) = \begin{cases} a x + b & x \u003c 0 \ x^2$ + 4 & x \geq 0 \end{cases}. $$For which values of $$a$$ and $$b is$$ $$f(x)$$ continuous at $$x = 0$$?

Accepted Answer

$$a = 0, b = 4$$

Question 19

Find the derivative of $$f(x) = \frac{5}{x^2$ + 2x + 1}.$$

Accepted Answer

$$-\frac{10x + 10}{(x^2$ + 2x + 1)^2$}$$

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