Question 1

Given the graph in Image 1, what type of discontinuity is present at $$x = 6$$?

Accepted Answer

Jump discontinuity

Question 2

Evaluate the following limit: $$\lim_{x 	o 0} \frac{\sin 3x}{4x}.$$

Accepted Answer

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

Question 3

Which of the following best describes the end behavior of the function in Image 1 as $$x 	o \infty$$?

Accepted Answer

The function approaches a horizontal asymptote at $$y = 6.$$

Question 4

Given $$f(x) = \sqrt{x+1} - \sqrt{7}$$, evaluate $$\lim_{x 	o 6} \frac{\sqrt{x+1} - \sqrt{7}}{x-6}.$$

Accepted Answer

$$\frac{1}{2\sqrt{7}}$$

Question 5

For the function $$f(x) = |x|$$, as shown in Image 2, what is the value of $$\lim_{x 	o 0} \frac{|x|}{x}$$?

Accepted Answer

The limit does not exist.

Question 6

Which of the following statements about the derivative at $$x = 0$$ for the function in Image 2 is correct?

Accepted Answer

The derivative does not exist due to a cusp.

Question 7

Given $$f(x) = -2x^2$ + 9x - 7$$, use the limit definition of the derivative to find $$f'(x). $$Which is correct?

Accepted Answer

$$f'(x) = -4x + 9$$

Question 8

Evaluate $$\lim_{x 	o 0} \frac{1 - \cos x}{x}.$$

Accepted Answer

$$0$$

Question 9

For the function $$f(x) = \frac{1}{x-3}$$, what is the derivative $$f'(x)$$ using the limit definition?

Accepted Answer

$$-\frac{1}{(x-3)^2$}$$

Question 10

Which of the following best describes the difference between an average rate of change and an instantaneous rate of change?

Accepted Answer

Average rate of change is the slope between two points, while instantaneous rate of change is the slope at a single point.

Question 11

Given $$f(x) = x^3$ - 5x^2$, find f$$'(x)$$ using the limit definition of the derivative.

Accepted Answer

$$3x^2 - 10x$

Question 12

For the function $$f(x) = \frac{1}{x}$$, what is the second derivative $$f''(x)$$ using the limit definition?

Accepted Answer

$$\frac{2}{x^3$}$$

Question 13

Evaluate $$\lim_{x 	o \infty} \frac{x^2$ + 5x + 6}{x^3$ - 2x^2$ - 15x}.$$

Accepted Answer

$$0$$

Question 14

Given the graph in Image 1, what is $$\lim_{x 	o 10^-} f(x)$$?

Accepted Answer

$$-\infty$$

Question 15

Which of the following is true about $$\lim_{x 	o \infty} |\sin x|$$?

Accepted Answer

The limit does not exist because $$|\sin x|$$ oscillates between $$0$$ and $$1.$$

Question 16

Given $$f(x) = \sqrt{x-1}$$, what is $$f'(x)$$ using the limit definition of the derivative?

Accepted Answer

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

Question 17

For the function $$f(x) = 3x^2$, what is f$$'(x)$$ using the limit definition of the derivative?

Accepted Answer

$$6x$$

Question 18

Given the graph in Image 3, what is the slope of the tangent line at $$x = 6$$?

Accepted Answer

$$\frac{1}{4}$$

Question 19

Evaluate $$\lim_{x 	o 0} \ln|x|.$$

Accepted Answer

$$-\infty$$

Question 20

Given $$f(x) = \begin{cases} x+2, & x  4 \end{cases}$$, what is $$\lim_{x 	o 4} f(x)$$?

Accepted Answer

The limit does not exist.

Step-by-Step Guidance for Calculus Exam Review: Limits and Derivatives

Study Guide - Practice Questions