$$\sin \theta = \dfrac{\text{opposite}}{\text{hypotenuse}}$$

$$\theta = \dfrac{\pi}{4}, \dfrac{5\pi}{4}$$

Question 1

Which of the following is the correct definition of $$\sin 	heta$$ for a right triangle?

Accepted Answer

$$\sin 	heta = \dfrac{	ext{opposite}}{	ext{hypotenuse}}$$

Question 2

Find all values of $$	heta in$$ $$[0, 2\pi]$$ that satisfy $$\sin 	heta = \cos 	heta.$$

Accepted Answer

$$	heta = \dfrac{\pi}{4}, \dfrac{5\pi}{4}$$

Question 3

Evaluate $$\displaystyle \lim_{x 	o 8} (\sqrt[3]{x} - x).$$

Accepted Answer

$$-6$$

Question 4

Which of the following best describes the behavior of $$\displaystyle \lim_{x 	o \infty} \dfrac{3x+2}{-5x+4}$$?

Accepted Answer

$$-\dfrac{3}{5}$$

Question 5

A company’s cost function is $$C(x) = 100 + 16x. $$What is the average cost function $$\overline{C}(x)$$?

Accepted Answer

$$\overline{C}(x) = \dfrac{100}{x} + 16$$

Question 6

Evaluate $$\displaystyle \lim_{x 	o 0} \dfrac{e$$^{2x}$$ - 1}{e^x - 1}.$$

Accepted Answer

$$2$$

Question 7

For the piecewise function $$f(x) = \begin{cases} 0 & x  0 \end{cases}$$, does $$\displaystyle \lim_{x 	o 0} f(x)$$ exist?

Accepted Answer

No, the left and right limits are not equal

Question 8

Evaluate $$\displaystyle \lim_{x 	o 1} \dfrac{\ln x^2$}{\ln x}.$$

Accepted Answer

$$2$$

Question 9

Evaluate $$\displaystyle \lim_{x 	o 0} \dfrac{\sin^2$(2x)}{\sin^2$(x)}.$$

Accepted Answer

$$4$$

Question 10

Evaluate $$\displaystyle \lim_{x 	o \infty} (\ln(x+1) - \ln x).$$

Accepted Answer

$$0$$

Question 11

For the function $$g(x) = \begin{cases} x^2$ & x  0 \end{cases}$$, does $$\displaystyle \lim_{x 	o 0} g(x)$$ exist?

Accepted Answer

Yes, the limit is $$0$$

Question 12

Evaluate $$\displaystyle \lim_{x 	o -\infty} \dfrac{|x|}{x}.$$

Accepted Answer

$$-1$$

Question 13

Evaluate $$\displaystyle \lim_{x 	o 4} \begin{cases} 0 & x \leq 0 \ x+1 & 0  4 \end{cases}.$$

Accepted Answer

$$5$$

Question 14

Evaluate $$\displaystyle \lim_{x 	o 1} \dfrac{\sqrt{x} - 1}{x - 1}.$$

Accepted Answer

$$\dfrac{1}{2}$$

Question 15

Evaluate $$\displaystyle \lim_{x 	o -\infty} \dfrac{-3x^3$ + 2x^2$ - 10x + 33}{1}.$$

Accepted Answer

$$\infty$$

Question 16

Evaluate $$\displaystyle \lim_{x 	o -\infty} \dfrac{-3x^4$ + 2x^2$ - 10x + 33}{1}.$$

Accepted Answer

$$-\infty$$

Question 17

Evaluate $$\displaystyle \lim_{x 	o -\infty} \dfrac{3x^4$ + 5x^3$ - 2x^2$ - 11x + 88}{-2x^4$ + 1}.$$

Accepted Answer

$$-\dfrac{3}{2}$$

Question 18

Evaluate $$\displaystyle \lim_{x 	o 2} \dfrac{x+2}{x^2$ - x - 6}.$$

Accepted Answer

$$-2$$

Question 19

Evaluate $$\displaystyle \lim_{x 	o 1} \dfrac{12x}{(x-1)^3$}.$$

Accepted Answer

$$\infty$$

Question 20

Evaluate $$\displaystyle \lim_{x 	o 1} \dfrac{-x}{(x-1)^2$}.$$

Accepted Answer

$$-\infty$$

Business Calculus Worksheet: Trigonometric Equations, Limits, and Cost Functions

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