Q: Suppose $$f(x) is $$defined everywhere except at $$x = 2. If$$ $$\lim_{x \to 2^-} f(x) = 5$$ and $$\lim_{x \to 2^+} f(x) = 3$$, what can you conclude about $$\lim_{x \to 2} f(x)$$?

$$\lim_{x \to 2} f(x)$$ does not exist

Question 1

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

Accepted Answer

As $$x$$ approaches $$a$$, the value of $$f(x)$$ can be made arbitrarily close to $$L by $$taking $$x$$ sufficiently close to $$a$$, but not equal to $$a.$$

Question 2

Suppose $$f(x) is $$defined everywhere except at $$x = 2. If$$ $$\lim_{x 	o 2^-} f(x) = 5$$ and $$\lim_{x 	o 2^+} f(x) = 3$$, what can you conclude about $$\lim_{x 	o 2} f(x)$$?

Accepted Answer

$$\lim_{x 	o 2} f(x)$$ does not exist

Question 3

Evaluate $$\lim_{x 	o \infty} \frac{2x^3$ + 5x}{x^2$ - 4}.$$

Accepted Answer

$$\infty$$

Question 4

A pharmaceutical company models the concentration of a drug in the bloodstream by $$C(t) = \frac{100}{t^2$ + 1}$$, where $$t is $$time in hours. What is $$\lim_{t 	o \infty} C(t)$$?

Accepted Answer

$$0$$

Question 5

Which of the following is TRUE about the limit $$\lim_{x 	o a} f(x)$$?

Accepted Answer

It depends only on the values of $$f(x) as$$ $$x$$ approaches $$a$$, not at $$a$$ itself.

Question 6

Evaluate $$\lim_{x 	o \infty} \frac{4x^2$ - 7x + 2}{2x^2$ + 5x - 3}.$$

Accepted Answer

$$2$$

Question 7

A company’s profit function is $$P(x) = \frac{5x^2$ + 3}{x^3$ - 2x}. $$What is $$\lim_{x 	o \infty} P(x)$$?

Accepted Answer

$$0$$

Question 8

If $$\lim_{x 	o 4} f(x) = 2$$ and $$\lim_{x 	o 4} g(x) = -3$$, what is $$\lim_{x 	o 4} [f(x) \cdot g(x)]$$?

Accepted Answer

$$-6$$

Question 9

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

Accepted Answer

$$\frac{1}{2}$$

Question 10

Which of the following best describes an infinite limit?

Accepted Answer

The function becomes arbitrarily large in magnitude as $$x$$ approaches $$a.$$

Question 11

Evaluate $$\lim_{x 	o \infty} \sqrt{x^2$ + 3x} - x.$$

Accepted Answer

Question 12

Suppose $$\lim_{x 	o 5^-} f(x) = \infty$$ and $$\lim_{x 	o 5^+} f(x) = -\infty. $$What does this indicate about the graph of $$f(x) at$$ $$x = 5$$?

Accepted Answer

There is a vertical asymptote at $$x = 5$$ with $$f(x)$$ approaching $$\infty$$ from the left and $$-\infty$$ from the right.

Question 13

If $$\lim_{x 	o 3} f(x) = 7$$ and $$\lim_{x 	o 3} g(x) = 2$$, what is $$\lim_{x 	o 3} \left( \frac{f(x)}{g(x)} ight)$$?

Accepted Answer

$$\frac{7}{2}$$

Question 14

A business models its cost function as $$C(x) = 10 + 2x. $$What is $$\lim_{x 	o 100} C(x)$$?

Accepted Answer

$$210$$

Question 15

Which property of limits allows you to compute $$\lim_{x 	o a} [f(x) + g(x)] as$$ $$\lim_{x 	o a} f(x) + \lim_{x 	o a} g(x)$$?

Accepted Answer

Sum property

Question 16

Evaluate $$\lim_{x 	o \infty} \frac{5x^3$ - 2x^2$ + 10}{-2x^3$ + x + 1}.$$

Accepted Answer

$$-\frac{5}{2}$$

Question 17

If $$\lim_{x 	o a} f(x) = 0$$ and $$\lim_{x 	o a} g(x) = 0$$, what should you do to evaluate $$\lim_{x 	o a} \frac{f(x)}{g(x)}$$?

Accepted Answer

Manipulate the expression algebraically to remove the $$\frac{0}{0}$$ form.

Question 18

A company’s revenue function is $$R(x) = \frac{3x^2$ + 5}{x^3$ - x + 1}. $$What is $$\lim_{x 	o \infty} R(x)$$?

Accepted Answer

$$0$$

Limits: Foundations and Properties in Business Calculus

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