Question 1

Given the function $$h(x)$$, which of the following statements is true if $$\lim_{x 	o 1^+} h(x) = \infty$$?

Accepted Answer

As $$x$$ approaches 1 from the right, $$h(x)$$ increases without bound.

Question 2

Evaluate the following limit: $$\lim_{x 	o \infty} \frac{3x + 2}{2x^2$ - 8}.$$

Accepted Answer

$$0$$

Question 3

Evaluate the following limit: $$\lim_{x 	o \infty} \frac{2x}{5 - 3x}.$$

Accepted Answer

$$-\frac{2}{3}$$

Question 4

Which of the following best describes the end behavior of a function $$f(x) if$$ $$\lim_{x 	o \infty} f(x) = 0$$?

Accepted Answer

As $$x$$ increases without bound, $$f(x)$$ approaches 0.

Question 5

Given $$h(x)$$ such that $$\lim_{x 	o -\infty} h(x) = 3$$, what does this mean about the graph of $$h(x)$$?

Accepted Answer

As $$x$$ decreases without bound, $$h(x)$$ approaches 3.

Question 6

Consider the function $$g(x) = \frac{x^2$ - 9}{2x^2$ - 2x - 12}. $$Which of the following statements about its continuity is correct?

Accepted Answer

It has a removable discontinuity at $$x = 3$$ and a nonremovable discontinuity at $$x = -2.$$

Question 7

Which of the following is a function with a removable discontinuity at $$x = 2$$ and a nonremovable discontinuity at $$x = -4$$?

Accepted Answer

$$f(x) = \frac{(x-2)}{x+4}$$

Question 8

A chemical reaction's rate $$R(t) is $$modeled by $$R(t) = \frac{5t}{t^2$ + 1}. $$What is $$\lim_{t 	o \infty} R(t)$$ and what does it mean physically?

Accepted Answer

$$0$$; the reaction rate approaches zero as time increases.

Question 9

Suppose $$f(x)$$ has a vertical asymptote at $$x = 1$$ and a horizontal asymptote at $$y = 0. $$Which of the following could be the equation for $$f(x)$$?

Accepted Answer

$$f(x) = \frac{1}{x-1}$$

Question 10

A medical dosage function is given by $$D(x) = \frac{10x}{x^2$ + 25}. $$What is the long-term effect of increasing $$x ($$the time since administration) on the dosage in the bloodstream?

Accepted Answer

The dosage approaches zero as time increases.

Question 11

Which of the following limits does NOT exist?

Accepted Answer

$$\lim_{x 	o 0} \frac{1}{x}$$

Question 12

If $$g(x) = \frac{x^2$ - 9}{2x^2$ - 2x - 12}$$, for which value of $$x$$ does $$g(x)$$ have a removable discontinuity?

Accepted Answer

$$x = 3$$

Question 13

A function $$f(x) is $$defined as $$f(x) = \frac{x^2$ - 4}{x - 2}. $$What type of discontinuity does $$f(x)$$ have at $$x = 2$$?

Accepted Answer

Removable discontinuity

Question 14

Which of the following best describes a nonremovable discontinuity?

Accepted Answer

A discontinuity that cannot be eliminated by redefining the function at a single point.

Question 15

Suppose $$h(x)$$ models the height of water in a tank over time, and $$\lim_{x 	o \infty} h(x) = 0. $$What does this mean in practical terms?

Accepted Answer

The water level approaches zero as time goes on.

Question 16

If $$f(x) = \frac{3x + 2}{2x^2$ - 8}$$, what is the horizontal asymptote of $$f(x)$$?

Accepted Answer

$$y = 0$$

Question 17

Which of the following functions has a vertical asymptote at $$x = -4$$?

Accepted Answer

$$f(x) = \frac{1}{x+4}$$

Question 18

A function $$f(x) is $$continuous everywhere except at $$x = 2$$, where it has a hole. Which of the following is most likely the form of $$f(x)$$?

Accepted Answer

$$f(x) = \frac{x^2$ - 4}{x - 2}$$

Question 19

If $$\lim_{x 	o 1^+} h(x) = \infty$$, which of the following statements is true about the graph of $$h(x) at$$ $$x = 1$$?

Accepted Answer

There is a vertical asymptote at $$x = 1.$$

Question 20

A function $$f(x) is $$defined as $$f(x) = \frac{x^2$ - 4}{x + 4}. $$What type of discontinuity does $$f(x)$$ have at $$x = -4$$?

Accepted Answer

Nonremovable discontinuity

Calculus Quiz 2 Guidance: Limits, Continuity, and Discontinuities

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