Q: Suppose the sequence $2.24, 2.249, 2.2499, 2.24999, \ldots$ is given. What is the limit of this sequence as $x$ increases, and how should it be written using limit notation?

$\lim\limits_{x \to 2.25^-} = 2.25$

Q: For the function $f(x) = \dfrac{x^2 - 9}{x - 3}$, what is $\lim\limits_{x \to 3} f(x)$?

$\lim\limits_{x \to 3} f(x) = 6$

Q: Consider the piecewise function $H(x) = \begin{cases} 2x+2 & \text{if } x<1 \\ 2x-4 & \text{if } x \geq 1 \end{cases}$. What is $\lim\limits_{x \to 1} H(x)$?

$\lim\limits_{x \to 1} H(x)$ does not exist

Q: For the same function $H(x)$ as above, what is $\lim\limits_{x \to -3} H(x)$?

$\lim\limits_{x \to -3} H(x) = -4$

Q: Given $f(x) = \dfrac{1}{x^2} + 3$, what is $\lim\limits_{x \to 3} f(x)$?

$\lim\limits_{x \to 3} f(x) = 4$

Q: For the function $f(x) = \dfrac{1}{x^2} + 3$, what is $\lim\limits_{x \to 0^+} f(x)$?

$\lim\limits_{x \to 0^+} f(x) = \infty$

Q: A function $f(x)$ is defined such that $\lim\limits_{x \to 2^-} f(x) = 5$ 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: Suppose $h(x) = \dfrac{1}{x-1} + 6$. What is $\lim\limits_{x \to 1} h(x)$?

$\lim\limits_{x \to 1} h(x)$ does not exist

Q: For the function $h(x) = \dfrac{1}{x-1} + 6$, what is $\lim\limits_{x \to 2} h(x)$?

$\lim\limits_{x \to 2} h(x) = 7$

Question 1

Suppose the sequence $2.24, 2.249, 2.2499, 2.24999, \ldots$ is given. What is the limit of this sequence as $x$ increases, and how should it be written using limit notation?

Accepted Answer

$\lim\limits_{x 	o 2.25^-} = 2.25$

Question 2

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

Accepted Answer

$f(3)$ does not exist$

Question 3

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

Accepted Answer

$\lim\limits_{x 	o 3} f(x) = 6$

Question 4

Which of the following statements is TRUE regarding the existence of $\lim\limits_{x 	o a} f(x)$?

Accepted Answer

The left-hand and right-hand limits must both exist and be equal.

Question 5

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

Accepted Answer

$\lim\limits_{x 	o 1} H(x)$ does not exist

Question 6

For the same function $H(x)$ as above, what is $\lim\limits_{x 	o -3} H(x)$?

Accepted Answer

$\lim\limits_{x 	o -3} H(x) = -4$

Question 7

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

Accepted Answer

$\lim\limits_{x 	o 3} f(x) = 4$

Question 8

For the function $f(x) = \dfrac{1}{x^2} + 3$, what is $\lim\limits_{x 	o 0^+} f(x)$?

Accepted Answer

$\lim\limits_{x 	o 0^+} f(x) = \infty$

Question 9

A function $f(x)$ is defined such that $\lim\limits_{x 	o 2^-} f(x) = 5$ 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 10

Suppose $h(x) = \dfrac{1}{x-1} + 6$. What is $\lim\limits_{x 	o 1} h(x)$?

Accepted Answer

$\lim\limits_{x 	o 1} h(x)$ does not exist

Question 11

For the function $h(x) = \dfrac{1}{x-1} + 6$, what is $\lim\limits_{x 	o 2} h(x)$?

Accepted Answer

$\lim\limits_{x 	o 2} h(x) = 7$

Question 12

Evaluate $\lim\limits_{x 	o \infty} \left( \dfrac{1}{x-1} + 6 ight)$. What is the result?

Accepted Answer

$\lim\limits_{x 	o \infty} h(x) = 6$

Question 13

A company’s cost function is $C(x) = \dfrac{1}{x^2} + 3$, where $x$ is the number of units produced. What happens to the cost per unit as production increases without bound?

Accepted Answer

The cost per unit approaches $3$

Question 14

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

Accepted Answer

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

Question 15

A function $f(x)$ is defined such that $f(2)$ does not exist, but $\lim\limits_{x 	o 2} f(x) = 5$. Which of the following is true?

Accepted Answer

The limit exists even though the function is undefined at $x=2$

Question 16

Which of the following best describes the 'wall method' for finding limits graphically?

Accepted Answer

Draw a vertical line at $x = a$ and observe the $y$-values approached from both sides.

Question 17

Suppose $f(x)$ is defined for all $x$ except $x = 4$, and $\lim\limits_{x 	o 4^-} f(x) = 2$, $\lim\limits_{x 	o 4^+} f(x) = 2$. What is $\lim\limits_{x 	o 4} f(x)$?

Accepted Answer

$\lim\limits_{x 	o 4} f(x) = 2$

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

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