Q: Which statement about one-sided limits is TRUE?

$\lim\limits_{x \to a^+} f(x)$ is the limit as $x$ approaches $a$ from the right.

Q: According to Theorem 2.1, when does $\lim\limits_{x \to a} f(x)$ exist?

If and only if $\lim\limits_{x \to a^-} f(x) = \lim\limits_{x \to a^+} f(x) = L$.

Q: Suppose $f(x)$ is not defined at $x=a$. Which of the following is TRUE?

$\lim\limits_{x \to a} f(x)$ can exist even if $f(a)$ is undefined.

Q: Using the graph of $y = f(x)$, you observe that as $x$ approaches $3$ from the left, $f(x)$ approaches $2$, and as $x$ approaches $3$ from the right, $f(x)$ approaches $4$. What can you conclude?

$\lim\limits_{x \to 3} f(x)$ does not exist.

Q: Which of the following statements is FALSE?

The value of $\lim\limits_{x \to a} f(x)$ is always found by computing $f(a)$.

Q: If $\lim\limits_{x \to a^-} f(x) = L_1$ and $\lim\limits_{x \to a^+} f(x) = L_2$ with $L_1 \neq L_2$, what can you say about $\lim\limits_{x \to a} f(x)$?

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

Question 1

Suppose $f(x)$ is defined for all $x$ near $a$ except possibly at $a$. Which of the following best describes $\lim\limits_{x 	o a} f(x) = L$?

Accepted Answer

As $x$ approaches $a$ from both sides, $f(x)$ gets arbitrarily close to $L$, regardless of $f(a)$.

Question 2

Given the function $f(x) = \frac{x^2 - 8}{4x - 2}$, use the table of values near $x=2$ to estimate $\lim\limits_{x 	o 2} f(x)$. Which of the following is closest to the correct value?

Accepted Answer

$2.5$

Question 3

Which statement about one-sided limits is TRUE?

Accepted Answer

$\lim\limits_{x 	o a^+} f(x)$ is the limit as $x$ approaches $a$ from the right.

Question 4

According to Theorem 2.1, when does $\lim\limits_{x 	o a} f(x)$ exist?

Accepted Answer

If and only if $\lim\limits_{x 	o a^-} f(x) = \lim\limits_{x 	o a^+} f(x) = L$.

Question 5

Given the graph of $f(x)$ below, $f(2)$ is defined and equals $3$, but $\lim\limits_{x 	o 2} f(x) = 5$. What does this illustrate?

Accepted Answer

The value of $f(2)$ does not affect the value of $\lim\limits_{x 	o 2} f(x)$.

Question 6

Evaluate $\lim\limits_{x 	o 1} \frac{\sqrt{x} - 1}{x - 1}$ using a table of values. What is the correct value?

Accepted Answer

$\frac{1}{2}$

Question 7

A function $f(x)$ is defined as $f(x) = \begin{cases} x^2 + 1 & 	ext{if } x \leq -1 \ 3 & 	ext{if } x > -1 \end{cases}$. What is $\lim\limits_{x 	o -1} f(x)$?

Accepted Answer

The limit does not exist.

Question 8

Suppose $f(x)$ is not defined at $x=a$. Which of the following is TRUE?

Accepted Answer

$\lim\limits_{x 	o a} f(x)$ can exist even if $f(a)$ is undefined.

Question 9

Using the graph of $y = f(x)$, you observe that as $x$ approaches $3$ from the left, $f(x)$ approaches $2$, and as $x$ approaches $3$ from the right, $f(x)$ approaches $4$. What can you conclude?

Accepted Answer

$\lim\limits_{x 	o 3} f(x)$ does not exist.

Question 10

Evaluate $\lim\limits_{h 	o 0} \frac{\sin h}{h}$ using your knowledge of limits and the unit circle.

Accepted Answer

$1$

Question 11

Which of the following statements is FALSE?

Accepted Answer

The value of $\lim\limits_{x 	o a} f(x)$ is always found by computing $f(a)$.

Question 12

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

Accepted Answer

$3$

Question 13

Suppose $f(x)$ is defined for all $x$ near $a$ except at $a$, and $\lim\limits_{x 	o a} f(x)$ exists. Which of the following must be true?

Accepted Answer

As $x$ approaches $a$, $f(x)$ approaches a single value from both sides.

Question 14

If $\lim\limits_{x 	o a^-} f(x) = L_1$ and $\lim\limits_{x 	o a^+} f(x) = L_2$ with $L_1 
eq L_2$, what can you say about $\lim\limits_{x 	o a} f(x)$?

Accepted Answer

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

Question 15

Which of the following is a real-world example where a limit is used to analyze a physical phenomenon?

Accepted Answer

Calculating the instantaneous speed of a car by finding the limit of average speed as the time interval approaches zero.

Question 16

Evaluate $\lim\limits_{x 	o 0} \cot x$.

Accepted Answer

The limit does not exist.

Question 17

Suppose $f(x)$ is defined as $f(x) = \frac{x^2 - 9}{x - 3}$. What is $\lim\limits_{x 	o 3} f(x)$?

Accepted Answer

$6$

Question 18

Which of the following is the correct definition of a right-sided limit?

Accepted Answer

$\lim\limits_{x 	o a^+} f(x)$ is the limit as $x$ approaches $a$ from the right.

Definitions and Evaluation of Limits in Calculus

Study Guide - Practice Questions

Study Guide - Flashcards