Q: Suppose $\lim_{x \to 3} f(x) = \text{DNE}$. Which of the following could be a reason?

The left and right limits as $x \to 3$ are not equal.

Question 1

Which of the following best describes the meaning of $\lim_{x 	o a} f(x) = L$?

Accepted Answer

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

Question 2

Suppose $f(2) = 	ext{DNE}$ and $\lim_{x 	o 2} f(x) = 4$. Which of the following is true?

Accepted Answer

$f(x)$ approaches 4 as $x$ approaches 2, even though $f(2)$ is undefined.

Question 3

Given the graph of $g(x)$ with $g(2) = 6$ and $\lim_{x 	o 2} g(x) = 4$, what can you conclude?

Accepted Answer

$g(x)$ has a removable discontinuity at $x=2$.

Question 4

If $\lim_{x 	o 1^-} g(x) = 2$ and $\lim_{x 	o 1^+} g(x) = 3$, what is $\lim_{x 	o 1} g(x)$?

Accepted Answer

$	ext{DNE}$

Question 5

A function $f(x)$ is defined as follows: $f(x) = \frac{x^2 - 4}{x-2}$ for $x 
eq 2$. What is $\lim_{x 	o 2} f(x)$?

Accepted Answer

$4$

Question 6

Suppose $\lim_{x 	o 3} f(x) = 	ext{DNE}$. Which of the following could be a reason?

Accepted Answer

The left and right limits as $x 	o 3$ are not equal.

Question 7

Given the graph of $g(x)$ with $g(-1) = 3$ and $\lim_{x 	o -1} g(x) = 2$, which of the following statements is correct?

Accepted Answer

$g(x)$ has a jump discontinuity at $x = -1$.

Question 8

If $\lim_{x 	o 2^-} f(x) = 4$ and $\lim_{x 	o 2^+} f(x) = 1$, what is $\lim_{x 	o 2} f(x)$?

Accepted Answer

$	ext{DNE}$

Question 9

A patient’s medication concentration in the bloodstream is modeled by $C(t)$, where $t$ is time in hours. If $\lim_{t 	o 5} C(t) = 2$ mg/L, what does this mean in a medical context?

Accepted Answer

As time approaches $5$ hours, the concentration approaches $2$ mg/L, regardless of the value at $t=5$.

Question 10

Suppose $\lim_{x 	o 3} g(x) = 4$ and $g(3) = 2$. Which of the following is true?

Accepted Answer

$g(x)$ has a removable discontinuity at $x=3$.

Question 11

If $\lim_{x 	o 1^-} g(x) = 2$ and $\lim_{x 	o 1^+} g(x) = 2$, but $g(1) = 5$, what type of discontinuity does $g(x)$ have at $x=1$?

Accepted Answer

Removable discontinuity

Question 12

A chemical reaction rate $R(t)$ is undefined at $t=10$ seconds, but $\lim_{t 	o 10} R(t) = 7$. What does this imply about the reaction at $t=10$?

Accepted Answer

The reaction rate approaches $7$ as $t$ approaches $10$ seconds, even though it is undefined at $t=10$.

Question 13

Given $\lim_{x 	o 2^-} f(x) = 4$ and $\lim_{x 	o 2^+} f(x) = 4$, but $f(2)$ is undefined, what is $\lim_{x 	o 2} f(x)$?

Accepted Answer

$4$

Question 14

If $\lim_{x 	o 3^-} g(x) = 4$ and $\lim_{x 	o 3^+} g(x) = 5$, what is $\lim_{x 	o 3} g(x)$?

Accepted Answer

$	ext{DNE}$

Question 15

A function $h(x)$ is defined for all $x$ except $x=1$. If $\lim_{x 	o 1} h(x)$ exists, which of the following must be true?

Accepted Answer

$h(x)$ approaches a single value as $x$ approaches $1$ from both sides.

Question 16

Suppose $\lim_{x 	o 2} f(x) = 4$ and $\lim_{x 	o 2} g(x) = 5$. What is $\lim_{x 	o 2} [f(x) + g(x)]$?

Accepted Answer

$9$

Question 17

If $\lim_{x 	o 1} g(x) = 2$ and $g(1) = 5$, which of the following is true?

Accepted Answer

$g(x)$ has a removable discontinuity at $x=1$.

Question 18

A function $f(x)$ is defined as $f(x) = \begin{cases} 2x+1 & x < 3 \ 7 & x = 3 \ x^2 & x > 3 \end{cases}$. What is $\lim_{x 	o 3} f(x)$?

Accepted Answer

$	ext{DNE}$

Question 19

Suppose $\lim_{x 	o 5^-} g(x) = 5$ and $\lim_{x 	o 5^+} g(x) = 5$. What is $\lim_{x 	o 5} g(x)$?

Accepted Answer

$5$

Definition of Limits and One-Sided Limits 2.2

Study Guide - Practice Questions

Study Guide - Flashcards