Q: Given $H(x)$ as above, does $\lim\limits_{x \to 1} H(x)$ exist?

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

Q: If $\lim\limits_{x \to 2^-} f(x) = 5$ and $\lim\limits_{x \to 2^+} f(x) = 7$, 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 defined everywhere except at $x=4$. If $\lim\limits_{x \to 4} f(x) = 10$, which of the following must be true?

As $x$ approaches $4$, $f(x)$ gets arbitrarily close to $10$.

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

$\lim\limits_{x \to a} f(x) = L$

Q: Suppose $f(x)$ is defined everywhere, but $f(a)$ is not defined. If $\lim\limits_{x \to a} f(x)$ exists, what does this mean?

The limit can exist even if $f(a)$ is undefined.

Q: Which of the following is true about the existence of a limit at $x=a$?

The left-hand and right-hand limits at $x=a$ must be equal.

Question 1

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

Accepted Answer

As $x$ approaches $a$, $f(x)$ becomes arbitrarily close to $L$, but $x$ does not have to equal $a$.

Question 2

Given the function $f(x) = \dfrac{x^2 - 9}{x - 3}$, what is $\lim\limits_{x 	o 3} f(x)$?

Accepted Answer

$6$

Question 3

For the function $f(x) = \dfrac{1}{x^2} + 3$, what is $\lim\limits_{x 	o \infty} f(x)$?

Accepted Answer

$3$

Question 4

Suppose $H(x) = \begin{cases} 2x+2 & 	ext{if } x<1 \ 2x-4 & 	ext{if } x \geq 1 \end{cases}$. What is $\lim\limits_{x 	o 1^-} H(x)$?

Accepted Answer

$4$

Question 5

Suppose $H(x) = \begin{cases} 2x+2 & 	ext{if } x<1 \ 2x-4 & 	ext{if } x \geq 1 \end{cases}$. What is $\lim\limits_{x 	o 1^+} H(x)$?

Accepted Answer

$-2$

Question 6

Given $H(x)$ as above, does $\lim\limits_{x 	o 1} H(x)$ exist?

Accepted Answer

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

Question 7

A chemical reaction's temperature $T$ (in $^0$C) is modeled by $T(x) = \dfrac{1}{x^2} + 3$, where $x$ is time in minutes. What is the long-term temperature as $x 	o \infty$?

Accepted Answer

$3$

Question 8

If $\lim\limits_{x 	o 2^-} f(x) = 5$ and $\lim\limits_{x 	o 2^+} f(x) = 7$, 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 9

A function $f(x)$ is defined everywhere except at $x=4$. If $\lim\limits_{x 	o 4} f(x) = 10$, which of the following must be true?

Accepted Answer

As $x$ approaches $4$, $f(x)$ gets arbitrarily close to $10$.

Question 10

Consider the sequence $5.51, 5.501, 5.5001, 5.50001, \ldots$. Which notation best describes the limit of this sequence?

Accepted Answer

$x 	o 5.5^+$

Question 11

Suppose $f(x)$ is defined as $f(x) = \dfrac{1}{x^2} + 3$. What is $\lim\limits_{x 	o 0^+} f(x)$?

Accepted Answer

$\infty$

Question 12

A business models its profit $P(x)$ as $P(x) = \dfrac{x^2 - 9}{x - 3}$. What is the profit as $x$ approaches $3$?

Accepted Answer

$6$

Question 13

If $\lim\limits_{x 	o a^-} f(x) = L$ and $\lim\limits_{x 	o a^+} f(x) = L$, what can you conclude?

Accepted Answer

$\lim\limits_{x 	o a} f(x) = L$

Question 14

A function $h(x)$ has $\lim\limits_{x 	o 2^-} h(x) = 5$ and $\lim\limits_{x 	o 2^+} h(x) = 5$. What is $\lim\limits_{x 	o 2} h(x)$?

Accepted Answer

$5$

Question 15

Suppose $f(x)$ is defined everywhere, but $f(a)$ is not defined. If $\lim\limits_{x 	o a} f(x)$ exists, what does this mean?

Accepted Answer

The limit can exist even if $f(a)$ is undefined.

Question 16

A medical dosage function $D(x)$ is defined as $D(x) = \dfrac{1}{x^2} + 3$. What happens to $D(x)$ as $x$ approaches $0$ from the right?

Accepted Answer

$D(x)$ approaches $\infty$

Question 17

If a function $f(x)$ has $\lim\limits_{x 	o 2^-} f(x) = 3$ and $\lim\limits_{x 	o 2^+} f(x) = 3$, but $f(2) = 5$, what is $\lim\limits_{x 	o 2} f(x)$?

Accepted Answer

$3$

Question 18

Which of the following is true about the existence of a limit at $x=a$?

Accepted Answer

The left-hand and right-hand limits at $x=a$ must be equal.

Question 19

A function $f(x)$ is given by $f(x) = \dfrac{1}{x^2} + 3$. What is $\lim\limits_{x 	o -\infty} f(x)$?

Accepted Answer

$3$

Limits: A Numerical and Graphical Approach (Business Calculus Study Notes)

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