Q: Evaluate the following limit: $\lim\limits_{x \to 36} \frac{36-x}{6-\sqrt{x}}$

$\lim\limits_{x \to 36} \frac{36-x}{6-\sqrt{x}} = 12$

Q: Evaluate the following limit: $\lim\limits_{x \to 2} \frac{x^3 - 2x^2}{x-2}$

$\lim\limits_{x \to 2} \frac{x^2(x-2)}{x-2} = \lim\limits_{x \to 2} x^2 = 4$

Q: Given the piecewise function $f(x) = \begin{cases} -7 & -8 \leq x < -3 \\ 2 - x^2 & -3 \leq x < 1 \\ -3x + 1 & x \geq 1 \end{cases}$, find $\lim\limits_{x \to -3} f(x)$.

$\lim\limits_{x \to -3^-} f(x) = -7$, $\lim\limits_{x \to -3^+} f(x) = 2 - (-3)^2 = -7$, \text{so the limit exists and is } -7$

Q: A company’s profit function is $P(x) = -3x^2 + 12x - 5$. What is $\lim\limits_{x \to \infty} P(x)$ and what does it mean in a business context?

$-\infty$. As production increases without bound, profit decreases without bound.

Question 1

Evaluate the following limit: $\lim\limits_{x 	o -4} (-3x^2 - 7x + 5)$

Accepted Answer

$-3(-4)^2 - 7(-4) + 5 = -48 + 28 + 5 = -15$

Question 2

Evaluate the following limit: $\lim\limits_{x 	o -2} \frac{x+2}{x^2 + 7x + 10}$

Accepted Answer

$\frac{0}{(-2)^2 + 7(-2) + 10} = \frac{0}{4 - 14 + 10} = \frac{0}{0}$ (undefined)

Question 3

Evaluate the following limit: $\lim\limits_{x 	o 36} \frac{36-x}{6-\sqrt{x}}$

Accepted Answer

$\lim\limits_{x 	o 36} \frac{36-x}{6-\sqrt{x}} = 12$

Question 4

Evaluate the following limit: $\lim\limits_{x 	o 2} \frac{x^3 - 2x^2}{x-2}$

Accepted Answer

$\lim\limits_{x 	o 2} \frac{x^2(x-2)}{x-2} = \lim\limits_{x 	o 2} x^2 = 4$

Question 5

Given the piecewise function $f(x) = \begin{cases} -7 & -8 \leq x < -3 \ 2 - x^2 & -3 \leq x < 1 \ -3x + 1 & x \geq 1 \end{cases}$, find $\lim\limits_{x 	o -3} f(x)$.

Accepted Answer

$\lim\limits_{x 	o -3^-} f(x) = -7$, $\lim\limits_{x 	o -3^+} f(x) = 2 - (-3)^2 = -7$, 	ext{so the limit exists and is } -7$

Question 6

Find $\lim\limits_{x 	o \infty} f(x)$ for $f(x) = 2x - 5$.

Accepted Answer

$\infty$

Question 7

Find $\lim\limits_{x 	o -\infty} f(x)$ for $f(x) = x^2$.

Accepted Answer

$\infty$

Question 8

Find $\lim\limits_{x 	o \infty} f(x)$ for $f(x) = \frac{3x^2 - 7}{4x^5 + 2x + 6}$.

Accepted Answer

$0$

Question 9

Find $\lim\limits_{x 	o \infty} f(x)$ for $f(x) = \frac{-5x^7 + 7x^4}{x^3 - 16}$.

Accepted Answer

$-\infty$

Question 10

Using the graph provided, what is $\lim\limits_{x 	o 5^-} f(x)$?

Accepted Answer

$\infty$

Question 11

Using the graph provided, what is $\lim\limits_{x 	o 5^+} f(x)$?

Accepted Answer

$-\infty$

Question 12

Using the graph provided, what is $\lim\limits_{x 	o -2^-} f(x)$?

Accepted Answer

$6$

Question 13

Using the graph provided, what is $\lim\limits_{x 	o -2^+} f(x)$?

Accepted Answer

$2$

Question 14

Using the graph provided, what is $f(-2)$?

Accepted Answer

$4$

Question 15

Using the graph provided, what is $\lim\limits_{x 	o 7^-} f(x)$?

Accepted Answer

$3$

Question 16

Using the graph provided, what is $f(7)$?

Accepted Answer

$4$

Question 17

A company’s profit function is $P(x) = -3x^2 + 12x - 5$. What is $\lim\limits_{x 	o \infty} P(x)$ and what does it mean in a business context?

Accepted Answer

$-\infty$. As production increases without bound, profit decreases without bound.

Business Calculus: Limits Practice and Graphical Analysis

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