Q: If $\lim\limits_{x \to 2^-} f(x) = 1$ and $\lim\limits_{x \to 2^+} f(x) = 3$, what can you conclude about $\lim\limits_{x \to 2} f(x)$?

$\lim\limits_{x \to 2} f(x)$ does not exist

Q: A function $f(x)$ is continuous at $x = a$ if:

$\lim\limits_{x \to a} f(x)$ exists and equals $f(a)$

Q: Given the piecewise function $f(x) = \begin{cases} x+1, & x < 2 \\ 2x-1, & x \geq 2 \end{cases}$, is $f(x)$ continuous at $x = 2$?

Yes, because $\lim\limits_{x \to 2^-} f(x) = \lim\limits_{x \to 2^+} f(x) = f(2)$

Q: Which of the following limit laws is correct?

$\lim\limits_{x \to a} [f(x) + g(x)] = \lim\limits_{x \to a} f(x) + \lim\limits_{x \to a} g(x)$

Q: A business analyst models the cost function $C(x)$ for producing $x$ units. If $\lim\limits_{x \to 100} C(x) = 5000$, what does this mean in practical terms?

As the number of units produced gets very close to 100, the cost approaches $5000.

Q: If $\lim\limits_{x \to 3} f(x) = 2$ and $f(3) = 5$, what can you say about the continuity of $f(x)$ at $x = 3$?

$f(x)$ is not continuous at $x = 3$.

Q: Which of the following is NOT a requirement for a function $f(x)$ to be continuous at $x = a$?

$f(x)$ is differentiable at $x = a$

Question 1

Which of the following statements best describes the limit of a function $f(x)$ as $x$ approaches $a$?

Accepted Answer

It is the value that $f(x)$ gets arbitrarily close to as $x$ gets close to $a$, but not necessarily equal to $a$.

Question 2

Given $f(x) = \frac{2x^2 - x - 1}{x - 1}$, what is $\lim\limits_{x 	o 1} f(x)$?

Accepted Answer

$3$

Question 3

If $\lim\limits_{x 	o 2^-} f(x) = 1$ and $\lim\limits_{x 	o 2^+} f(x) = 3$, what can you conclude about $\lim\limits_{x 	o 2} f(x)$?

Accepted Answer

$\lim\limits_{x 	o 2} f(x)$ does not exist

Question 4

A function $f(x)$ is continuous at $x = a$ if:

Accepted Answer

$\lim\limits_{x 	o a} f(x)$ exists and equals $f(a)$

Question 5

Which of the following is an example of a jump discontinuity?

Accepted Answer

The graph of $f(x)$ has a sudden jump from one value to another at $x = c$.

Question 6

Evaluate $\lim\limits_{x 	o 0^-} f(x)$ and $\lim\limits_{x 	o 0^+} f(x)$ for a function where $f(x)$ approaches $2$ from the left and $4$ from the right as $x$ approaches $0$. What is $\lim\limits_{x 	o 0} f(x)$?

Accepted Answer

Does not exist

Question 7

Suppose $f(x)$ is defined as $f(x) = \sqrt{x^2 - 4x}$ for all $x \geq 4$. What is $\lim\limits_{x 	o 4} f(x)$?

Accepted Answer

$0$

Question 8

Which of the following is true about the left-hand and right-hand limits?

Accepted Answer

If the left-hand and right-hand limits at $x = a$ are equal, then $\lim\limits_{x 	o a} f(x)$ exists.

Question 9

Given the piecewise function $f(x) = \begin{cases} x+1, & x < 2 \ 2x-1, & x \geq 2 \end{cases}$, is $f(x)$ continuous at $x = 2$?

Accepted Answer

Yes, because $\lim\limits_{x 	o 2^-} f(x) = \lim\limits_{x 	o 2^+} f(x) = f(2)$

Question 10

Which of the following limit laws is correct?

Accepted Answer

$\lim\limits_{x 	o a} [f(x) + g(x)] = \lim\limits_{x 	o a} f(x) + \lim\limits_{x 	o a} g(x)$

Question 11

A business analyst models the cost function $C(x)$ for producing $x$ units. If $\lim\limits_{x 	o 100} C(x) = 5000$, what does this mean in practical terms?

Accepted Answer

As the number of units produced gets very close to 100, the cost approaches $5000.

Question 12

If $\lim\limits_{x 	o 3} f(x) = 2$ and $f(3) = 5$, what can you say about the continuity of $f(x)$ at $x = 3$?

Accepted Answer

$f(x)$ is not continuous at $x = 3$.

Question 13

Which of the following is NOT a requirement for a function $f(x)$ to be continuous at $x = a$?

Accepted Answer

$f(x)$ is differentiable at $x = a$

Question 14

Evaluate $\lim\limits_{x 	o 2} \sqrt{x^2 - 4x + 4}$.

Accepted Answer

$0$

Question 15

A function $f(x)$ has a hole at $x = 5$. What does this mean?

Accepted Answer

$f(x)$ is not defined at $x = 5$, but $\lim\limits_{x 	o 5} f(x)$ exists.

Question 16

If $\lim\limits_{x 	o a} f(x) = L$ and $\lim\limits_{x 	o a} g(x) = M$, what is $\lim\limits_{x 	o a} [f(x) \cdot g(x)]$?

Accepted Answer

$L \cdot M$

Question 17

A company’s revenue function $R(x)$ is continuous at $x = 100$. What does this imply about $R(x)$ at $x = 100$?

Accepted Answer

$\lim\limits_{x 	o 100} R(x) = R(100)$

Business Calculus: Limits and Continuity (Section 1.1 Study Notes)

Study Guide - Practice Questions

Study Guide - Flashcards