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

A value $x = a$ where $\lim_{x \to a} f(x) = \infty$ or $-\infty$.

$\lim_{x \to a} [f(x)g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)$

Question 1

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

Accepted Answer

As $x$ approaches $a$, $f(x)$ becomes arbitrarily close to $L$.

Question 2

Given the function $f(x) = \frac{x^2 - 4}{x - 2}$, what is $\lim_{x 	o 2} f(x)$?

Accepted Answer

$4$

Question 3

If $\lim_{x 	o 3^-} f(x) = 5$ and $\lim_{x 	o 3^+} f(x) = 7$, what can you conclude about $\lim_{x 	o 3} f(x)$?

Accepted Answer

The limit does not exist.

Question 4

Evaluate $\lim_{x 	o \infty} \frac{3x^2 + 2}{5x^2 - 7}$.

Accepted Answer

$\frac{3}{5}$

Question 5

Which of the following is a correct application of the limit sum law?

Accepted Answer

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

Question 6

Suppose $\lim_{x 	o 1} f(x) = 2$ and $\lim_{x 	o 1} g(x) = -3$. What is $\lim_{x 	o 1} [2f(x) - 3g(x)]$?

Accepted Answer

$2 \cdot 2 - 3 \cdot (-3) = 4 + 9 = 13$

Question 7

A function $f(x)$ is defined as $f(x) = \frac{x^2 - 9}{x - 3}$. What is the value of $\lim_{x 	o 3} f(x)$?

Accepted Answer

$6$

Question 8

Which of the following best describes a vertical asymptote of a rational function $f(x)$?

Accepted Answer

A value $x = a$ where $\lim_{x 	o a} f(x) = \infty$ or $-\infty$.

Question 9

Evaluate $\lim_{x 	o 0} \frac{\sin x}{x}$.

Accepted Answer

$1$

Question 10

If $\lim_{x 	o 2} f(x) = 5$ and $\lim_{x 	o 2} g(x) = -1$, what is $\lim_{x 	o 2} [f(x)g(x)]$?

Accepted Answer

$-5$

Question 11

Given $f(x) = \frac{2x^2 + 3x - 2}{x^2 - 1}$, what is the horizontal asymptote as $x 	o \infty$?

Accepted Answer

$y = 2$

Question 12

Which of the following limits does NOT exist?

Accepted Answer

$\lim_{x 	o 0} \frac{1}{x}$

Question 13

Evaluate $\lim_{x 	o 1} \frac{x^2 - 1}{x - 1}$.

Accepted Answer

$2$

Question 14

Suppose $\lim_{x 	o 4} f(x) = 7$ and $\lim_{x 	o 4} g(x) = 2$. What is $\lim_{x 	o 4} \frac{f(x)}{g(x)}$?

Accepted Answer

$\frac{7}{2}$

Question 15

Which of the following is the correct limit law for the product of two functions?

Accepted Answer

$\lim_{x 	o a} [f(x)g(x)] = \lim_{x 	o a} f(x) \cdot \lim_{x 	o a} g(x)$

Question 16

Evaluate $\lim_{x 	o 0} \frac{e^x - 1}{x}$.

Accepted Answer

$1$

Question 17

A company’s profit function is $P(x) = \frac{100x}{x+10}$. What is the limiting profit as $x 	o \infty$?

Accepted Answer

$100$

Question 18

Which of the following is true about the limit $\lim_{x 	o a} f(x)$ if $f(x)$ has a hole at $x = a$?

Accepted Answer

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

Question 19

Evaluate $\lim_{x 	o 2} \frac{x^2 - 4}{x^2 + 2x - 8}$.

Accepted Answer

$0$

Limits and Their Applications in Business Calculus

Study Guide - Practice Questions