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 $$f(x) = \begin{cases} -7 & -8 \leq x \u003c -3 \\ 2 - x^2$ & -3 \leq x \u003c 1 \\ -3x + 1 & x \geq 1 \end{cases}$$, 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$

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 $$f(x) = \begin{cases} -7 & -8 \leq x \u003c -3 \ 2 - x^2$ & -3 \leq x \u003c 1 \ -3x + 1 & x \geq 1 \end{cases}$$, 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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