Question 1

Given the graph in Image 1, what type of discontinuity is present at $$x = 5$$?

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 $$y = 6.$$

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 $$f(x) = -2x^2$ + 9x - 7$$, what is the equation of the secant line through $$x = 2$$ and $$x = 5$$?

Accepted Answer

$$y = -5x + 13$$

Question 6

Using the limit definition of the derivative, what is $$f'(x)$$ for $$f(x) = 3x^2$?

Accepted Answer

$$6x$$

Question 7

Given the graph in Image 2, what is the value of the derivative at $$x = 0$$?

Accepted Answer

The derivative is undefined at $$x = 0.$$

Question 8

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

Accepted Answer

$$0$$

Question 9

What is the difference between a limit of $$\infty (e.g$$., $$\lim_{x 	o c} f(x) = \infty) $$and a limit at $$\infty (e.g$$., $$\lim_{x 	o \infty} f(x) = L$$)?

Accepted Answer

A limit of $$\infty$$ means the function grows without bound near a specific point, while a limit at $$\infty$$ describes the function's behavior as $$x$$ becomes very large.

Question 10

Find the second derivative of $$f(x) = \frac{1}{x}$$ using the limit definition.

Accepted Answer

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

Question 11

Given the graph in Image 3, estimate the slope of the tangent line at $$x = 6.$$

Accepted Answer

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

Question 12

For the piecewise function $$f(x) = \begin{cases} x+2, & x  4 \end{cases}$$, what is $$\lim_{x 	o 0} f(x)$$?

Accepted Answer

$$2$$

Question 13

Which of the following is an indeterminate form when evaluating limits?

Accepted Answer

$$\frac{0}{0}$$

Question 14

If $$\lim_{x 	o 0} |\csc x| is $$evaluated, what is the result?

Accepted Answer

$$\infty$$

Question 15

What is the average rate of change of $$f(x) = x^2$ from x = 1$ to x = 3$?

Accepted Answer

$$4$$

Question 16

Which strategy is most effective for simplifying a compound fraction when evaluating a limit?

Accepted Answer

Multiply numerator and denominator by the least common denominator of all inner fractions.

Question 17

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

Accepted Answer

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

Question 18

Given $$f(x) = x^3$ - 5x^2$, what is f$$'(x)$$ using the limit definition?

Accepted Answer

$$3x^2 - 10x$

Question 19

What is the correct notation for the derivative of $$f(x) at$$ $$x = a$$?

Accepted Answer

$$f'(a)$$

Question 20

If $$\lim_{x 	o \infty} |\sin x| is $$evaluated, what is the result?

Accepted Answer

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

MATH 10524 Exam I Review – Limits and Derivatives Study Guide

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