Q: Which of the following best describes the continuity of $$f(x) at$$ $$x = 2$$ based on the graph?

Not continuous, because $$\lim_{x \to 2^-} f(x) \neq \lim_{x \to 2^+} f(x)$$

Q: For the piecewise function $$h(x)$$ defined as $$h(x) = \begin{cases} x^2$ - 1 & x \u003c -3 \\ 3x + 17 & -3 \leq x \u003c 2 \\ -1 & x \geq 2 \end{cases}$$, is $$h(x)$$ continuous at $$x = -3$$?

Yes, because $$\lim_{x \to -3} h(x) = h(-3)$$

Question 1

Evaluate $$\lim_{x 	o -1^-} f(x)$$ using the graph provided.

Accepted Answer

$$-5$$

Question 2

Which of the following best describes the continuity of $$f(x) at$$ $$x = 2$$ based on the graph?

Accepted Answer

Not continuous, because $$\lim_{x 	o 2^-} f(x) 
eq \lim_{x 	o 2^+} f(x)$$

Question 3

Evaluate $$\lim_{x 	o 4} (3x - 2). $$Choose the correct answer.

Accepted Answer

$$10$$

Question 4

Find $$\lim_{x 	o 0} \frac{x + 3}{x - 2}.$$

Accepted Answer

$$-3$$

Question 5

Determine the horizontal asymptote of $$f(x) = \frac{3x^2$ - 5x + 1}{7x^2$ - 2x - 5}.$$

Accepted Answer

$$y = \frac{3}{7}$$

Question 6

For $$f(x) = \sqrt{3x - 5}$$, what is the interval of continuity?

Accepted Answer

$$[\frac{5}{3}, \infty)$$

Question 7

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

Accepted Answer

$$5$$

Question 8

Which of the following is the derivative of $$f(x) = x^2$ - 4x$$?

Accepted Answer

$$2x - 4$$

Question 9

A chemical reaction's rate is modeled by $$f(x) = 2x^2$ - 5x - 3. $$What is the vertical asymptote of this function?

Accepted Answer

$$x = -3$$

Question 10

Find the equation of the tangent line to $$f(x) = x^2$ + 4 at$$ $$x = -3.$$

Accepted Answer

$$y = -6x - 5$$

Question 11

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

Accepted Answer

$$0$$

Question 12

For the piecewise function $$h(x)$$ defined as $$h(x) = \begin{cases} x^2$ - 1 & x \u003c -3 \ 3x + 17 & -3 \leq x \u003c 2 \ -1 & x \geq 2 \end{cases}$$, is $$h(x)$$ continuous at $$x = -3$$?

Accepted Answer

Yes, because $$\lim_{x 	o -3} h(x) = h(-3)$$

Question 13

Evaluate $$\lim_{x 	o 4} \frac{x - 4}{\sqrt{x} - 2}.$$

Accepted Answer

$$4$$

Question 14

Which of the following is the correct interval of continuity for $$f(x) = \frac{2x - 1}{x^2$ + 2x - 8}$$?

Accepted Answer

$$( -\infty, -4 ) \cup ( -4, 2 ) \cup ( 2, \infty )$$

Question 15

A population model is given by $$f(x) = 5x^3$ - 2x + 6. $$What is $$\lim_{x 	o \infty} \frac{5x^3$ - 2x + 6}{3x^2$ + x - 5}$$?

Accepted Answer

$$\infty$$

Question 16

Evaluate $$\lim_{x 	o 1^-} \frac{x^2$ - x - 12}{x - 1}.$$

Accepted Answer

$$\infty$$

Question 17

Find the derivative of $$f(x) = x^2$ + 4$$ using the limit definition.

Accepted Answer

$$2x$$

Question 18

A medication's concentration is modeled by $$f(x) = x^2$ - 4x. $$What is the equation of the tangent line at $$x = 3$$?

Accepted Answer

$$y = 2x - 9$$

Question 19

Which of the following is the correct vertical asymptote for $$f(x) = \frac{x^2$ - 1}{x + 2}$$?

Accepted Answer

$$x = -2$$

Limits and Continuity from a Graph – Step-by-Step Study Guide

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