Q: Which of the following statements about limits is TRUE?

If $$\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)$$, then $$\lim_{x \to a} f(x)$$ exists.

Question 1

Evaluate the following limit, which models the instantaneous rate of change of the area of a square as its side length increases from 10 cm:

$$\lim_{h 	o 0} \frac{(10 + h)^2$ - 100}{h}$$

Accepted Answer

$$20$$

Question 2

A chemical reaction's rate is modeled by $$f(x) = \frac{x-2}{x^2$-2x}. $$What is $$\lim_{x 	o 2} f(x)$$?

Accepted Answer

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

Question 3

Evaluate the limit:

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

Accepted Answer

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

Question 4

A ball is thrown upward and its height in meters after $$t$$ seconds is $$h(t) = 3t^2$ - 4t. $$What is $$\lim_{t 	o 2} h(t)$$?

Accepted Answer

$$4$$

Question 5

Evaluate the following limit, which could represent the velocity of a car as time approaches 0:

$$\lim_{h 	o 0} \frac{\sqrt{49 + h} - 7}{h}$$

Accepted Answer

$$\frac{1}{14}$$

Question 6

Given the piecewise function

$$ f(x) = \begin{cases} -2x + 4 & 	ext{if } x \leq 1 \ \sqrt{x-1} & 	ext{if } x \u003e 1 \end{cases} $$

What is $$\lim_{x 	o 1^-} f(x)$$?

Accepted Answer

$$2$$

Question 7

Given the same function as above, what is $$\lim_{x 	o 1^+} f(x)$$?

Accepted Answer

$$0$$

Question 8

For the function $$f(x) = \frac{x-42}{\sqrt{x} - \sqrt{42}}$$, evaluate $$\lim_{x 	o 42} f(x)$$, which could represent a physical process as a parameter approaches a critical value.

Accepted Answer

$$2\sqrt{42}$$

Question 9

Suppose $$\lim_{x 	o 2} f(x) = 4$$, $$\lim_{x 	o 2} g(x) = 5$$, and $$\lim_{x 	o 2} h(x) = 8. $$What is $$\lim_{x 	o 2} \frac{f(x) - g(x)}{h(x)}$$?

Accepted Answer

$$\frac{1}{8}$$

Question 10

Evaluate the following limit, which could represent the change in concentration of a substance:

$$\lim_{x 	o 2} \frac{x^2$ - 6x + 8}{x^2$ - 4}$$

Accepted Answer

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

Question 11

A function is defined as $$f(x) = \frac{3x^2$ - 4x}{5x^3$ - 36}. $$What is $$\lim_{x 	o 2} f(x)$$?

Accepted Answer

$$1$$

Question 12

Given $$f(x) = 0 if$$ $$x \leq -4$$, $$f(x) = \sqrt{16 - x^2$} if$$ $$-4 \u003c x \u003c 4$$, and $$f(x) = 2x if$$ $$x \geq 4$$, what is $$\lim_{x 	o -4^-} f(x)$$?

Accepted Answer

$$0$$

Question 13

For the same function as above, what is $$\lim_{x 	o -4^+} f(x)$$?

Accepted Answer

$$0$$

Question 14

Suppose $$\lim_{x 	o 2} g(x) = 5. $$What is $$\lim_{x 	o 2} (g(x))^3$?

Accepted Answer

$$125$$

Question 15

If $$\lim_{x 	o 2} f(x) = 4$$, $$\lim_{x 	o 2} g(x) = 5$$, and $$\lim_{x 	o 2} h(x) = 8$$, what is $$\lim_{x 	o 2} (6f(x)g(x) + h(x))$$?

Accepted Answer

$$128$$

Question 16

Evaluate the following limit, which could represent the change in a population as a parameter approaches a threshold:

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

Accepted Answer

$$-1$$

Question 17

Which of the following statements about limits is TRUE?

Accepted Answer

If $$\lim_{x 	o a^-} f(x) = \lim_{x 	o a^+} f(x)$$, then $$\lim_{x 	o a} f(x)$$ exists.

Question 18

Evaluate the following limit, which could represent the marginal cost in economics:

$$\lim_{h 	o 0} \frac{(a + h)^2$ - a^2$}{h}$$

Accepted Answer

Computing Limits: Methods and Examples 2.3

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