Q: Is $k(x)$ continuous at $x=0$ based on the graph in #18?

No, because $\lim\limits_{x \to 0} k(x) \neq k(0)$.

Question 1

Given the piecewise function $g(x) = \begin{cases} x+2 & 	ext{for } x<2 \ \frac{1}{2}x+1 & 	ext{for } x \geq 2 \end{cases}$, what is $\lim\limits_{x 	o 2^-} g(x)$?

Accepted Answer

$3$

Question 2

For the same function $g(x)$ as above, what is $\lim\limits_{x 	o 2^+} g(x)$?

Accepted Answer

$2.5$

Question 3

Does $\lim\limits_{x 	o 2} g(x)$ exist for the function $g(x)$ defined above?

Accepted Answer

No, because the left and right limits are not equal.

Question 4

Given the graph of $F(x)$ in #2, what is $\lim\limits_{x 	o 1} F(x)$?

Accepted Answer

$-1$

Question 5

Given the graph of $F(x)$ in #3, what is $\lim\limits_{x 	o 4} F(x)$?

Accepted Answer

$-4$

Question 6

Suppose $h(x)$ is graphed as in #6. What is $\lim\limits_{x 	o 2} h(x)$?

Accepted Answer

$8$

Question 7

Given the graph in #7, what is $\lim\limits_{x 	o 0} m(x)$?

Accepted Answer

$3$

Question 8

Evaluate $\lim\limits_{x 	o 4} (6x + 1)$. Choose the correct answer.

Accepted Answer

$25$

Question 9

Find $\lim\limits_{x 	o 5} \frac{x^2 - 9}{5 - x}$.

Accepted Answer

$-8$

Question 10

Find $\lim\limits_{x 	o 10} \frac{x^2 - 100}{x - 10}$.

Accepted Answer

$20$

Question 11

Find $\lim\limits_{x 	o 5} \frac{x^2 + 3x - 40}{x^2 - 25}$.

Accepted Answer

$\frac{8}{5}$

Question 12

Given the function $g(x) = \begin{cases} x+8 & x \leq 9 \ 5x-30 & x > 9 \end{cases}$, what is $\lim\limits_{x 	o 9^-} g(x)$?

Accepted Answer

$17$

Question 13

For the same $g(x)$ as above, what is $\lim\limits_{x 	o 9^+} g(x)$?

Accepted Answer

$15$

Question 14

Does $\lim\limits_{x 	o 9} g(x)$ exist for the function above?

Accepted Answer

No, because the left and right limits are not equal.

Question 15

Suppose $k(x)$ is graphed as in #18. What is $\lim\limits_{x 	o 0} k(x)$?

Accepted Answer

$-3$

Question 16

Is $k(x)$ continuous at $x=0$ based on the graph in #18?

Accepted Answer

No, because $\lim\limits_{x 	o 0} k(x) 
eq k(0)$.

Question 17

Given the graph in #16, is the function continuous over the interval $(-6,6)$?

Accepted Answer

Yes, the function is continuous over $(-6,6)$.

Question 18

Given the graph in #17, is the function continuous over the interval $(-6,6)$?

Accepted Answer

No, because there is a jump discontinuity.

Question 19

Suppose $f(x)$ is defined as $f(x) = \begin{cases} x^2 & x < 2 \ k+2 & x \geq 2 \end{cases}$. For what value of $k$ is $f(x)$ continuous at $x=2$?

Accepted Answer

$4$

Question 20

A business's profit function is $P(x) = 6x + 1$. What is the instantaneous rate of change of profit as $x$ approaches $4$?

Accepted Answer

$6$

Limits and Continuity: Graphical and Numerical Approaches in Business Calculus

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