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

$$\lim_{x \to a} f(x)$$ does not exist.

Q: A company models its revenue with $$R(x) = \frac{1}{x}$$, where $$x is $$the number of units sold. What is $$\lim_{x \to \infty} R(x)$$ and what does it mean in business terms?

$$\lim_{x \to \infty} R(x) = 0$$, meaning revenue per unit decreases as sales increase.

Question 1

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

Accepted Answer

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

Question 2

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

Accepted Answer

$$4$$

Question 3

Consider the function $$g(x) = \begin{cases} 1 + 2x & 	ext{if } x \leq 1 \ 2 - x & 	ext{if } x \u003e 1 \end{cases}. $$What is $$\lim_{x 	o 1^-} g(x)$$?

Accepted Answer

$$3$$

Question 4

For the function $$g(x) in $$the previous question, what is $$\lim_{x 	o 1^+} g(x)$$?

Accepted Answer

$$1$$

Question 5

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

Accepted Answer

$$\lim_{x 	o a} f(x)$$ does not exist.

Question 6

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

Accepted Answer

$$\infty$$

Question 7

A business cost function is given by $$C(x) = \frac{x}{x^2$ + 3x - 10}. $$What happens to $$C(x) as$$ $$x$$ approaches $$-5$$ from the left?

Accepted Answer

$$C(x)$$ approaches $$-\infty$$

Question 8

Given $$f(x) = \frac{1}{x^2$}$$, what is $$\lim_{x 	o \infty} f(x)$$?

Accepted Answer

$$0$$

Question 9

Suppose $$f(x) is $$defined everywhere except at $$x = 3$$, and $$\lim_{x 	o 3} f(x) = 7. $$Which statement is true?

Accepted Answer

$$f(3)$$ can be any value or undefined, but the limit as $$x$$ approaches $$3 is$$ $$7$$

Question 10

Evaluate $$\lim_{x 	o 6} \frac{x^2$ - 11x + 30}{x - 6}.$$

Accepted Answer

$$7$$

Question 11

A company models its revenue with $$R(x) = \frac{1}{x}$$, where $$x is $$the number of units sold. What is $$\lim_{x 	o \infty} R(x)$$ and what does it mean in business terms?

Accepted Answer

$$\lim_{x 	o \infty} R(x) = 0$$, meaning revenue per unit decreases as sales increase.

Question 12

Given $$g(x) = \begin{cases} x^2$ - 1 & x \leq 1 \ x^2$ - 2x + 1 & x \u003e 1 \end{cases}$$, what is $$\lim_{x 	o 1} g(x)$$?

Accepted Answer

$$-1$$

Question 13

If $$f(x) = \frac{x^2$ + 13x + 30}{x^2$ + 10x + 21}$$, what is $$\lim_{x 	o -3} f(x)$$?

Accepted Answer

$$\frac{4}{3}$$

Question 14

Suppose $$f(x) = \frac{x^2$ - 4x - 12}{x^2$ - 36}. $$What is $$\lim_{x 	o 6} f(x)$$?

Accepted Answer

$$-1$$

Question 15

For the function $$g(x) = \frac{x - 2}{x^2$ - 4}$$ for $$x 
eq 2$$, and $$g(2) = \frac{1}{4}$$, what is $$\lim_{x 	o 2} g(x)$$?

Accepted Answer

$$0$$

Question 16

If $$f(x) = \frac{1}{x}$$, what is $$\lim_{x 	o 0^-} f(x)$$?

Accepted Answer

$$-\infty$$

Question 17

A company’s profit function is $$P(x) = \frac{x^2$ - x - 12}{x + 2}. $$What is $$\lim_{x 	o -2} P(x)$$?

Accepted Answer

$$-7$$

Question 18

If the graph of $$f(x)$$ shows that as $$x 	o \infty$$, $$f(x)$$ approaches $$2$$, what is $$\lim_{x 	o \infty} f(x)$$?

Accepted Answer

$$2$$

Question 19

Suppose $$f(x) is $$defined everywhere except at $$x = 0$$, and the graph shows $$\lim_{x 	o 0^-} f(x) = 3$$ and $$\lim_{x 	o 0^+} f(x) = 3. $$What is $$\lim_{x 	o 0} f(x)$$?

Accepted Answer

$$3$$

Limits and One-Sided Limits: Foundations for Business Calculus

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