$$f is $$continuous at $$a if$$ $$\lim_{x \to a} f(x) = f(a).$$

Yes, because $$\lim_{x \to 2^-} f(x) = \lim_{x \to 2^+} f(x) = f(2).$$

$$\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)$$, provided both limits exist.

The limit $$\lim_{x \to a} f(x)$$ exists, but $$f(a) is $$either undefined or not equal to the limit.

Question 1

Let $$f(x) = \frac{2x^2$ - 3x + 1}{x^2$ + x - 2}. $$What is the horizontal asymptote of the graph of $$f(x)$$?

Accepted Answer

$$y = 2$$

Question 2

Evaluate $$\lim_{x 	o 3^-} \frac{1}{x-3}.$$

Accepted Answer

$$-\infty$$

Question 3

Which of the following statements is TRUE about the function $$f(x) = 	an x$$?

Accepted Answer

It has vertical asymptotes at $$x = \frac{\pi}{2} + n\pi$$, where $$n is $$any integer.

Question 4

If $$\lim_{x 	o \infty} f(x) = 5$$, which of the following is TRUE?

Accepted Answer

The graph of $$f(x)$$ has a horizontal asymptote at $$y = 5.$$

Question 5

Which of the following is the correct definition of continuity at a point $$x = a$$?

Accepted Answer

$$f is $$continuous at $$a if$$ $$\lim_{x 	o a} f(x) = f(a).$$

Question 6

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

Accepted Answer

$$3$$

Question 7

A chemical reaction rate is modeled by $$R(t) = \frac{10}{t+2}. $$What is $$\lim_{t 	o \infty} R(t)$$ and what does it represent?

Accepted Answer

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

Question 8

Which of the following functions has a vertical asymptote at $$x = 1$$?

Accepted Answer

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

Question 9

Suppose $$f(x) is $$defined as $$f(x) = \begin{cases} x^2$ & x \u003c 2 \ 4x - 4 & x \geq 2 \end{cases}. Is$$ $$f(x)$$ continuous at $$x = 2$$?

Accepted Answer

Yes, because $$\lim_{x 	o 2^-} f(x) = \lim_{x 	o 2^+} f(x) = f(2).$$

Question 10

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

Accepted Answer

$$\infty$$

Question 11

A medication's concentration in the bloodstream is modeled by $$C(t) = \frac{50}{t^2$ + 1}. $$What is the horizontal asymptote of $$C(t)$$?

Accepted Answer

$$y = 0$$

Question 12

Which of the following is NOT a requirement for a function $$f(x) to be $$continuous at $$x = a$$?

Accepted Answer

$$f(a) is $$undefined.

Question 13

Evaluate $$\lim_{x 	o 2} \frac{x^2$ - 4}{x-2}.$$

Accepted Answer

$$4$$

Question 14

Which of the following best describes the behavior of $$f(x) = \frac{5x^3$ + 2}{2x^3$ - 7} as$$ $$x 	o \infty$$?

Accepted Answer

$$f(x)$$ approaches $$\frac{5}{2}.$$

Question 15

If $$\lim_{x 	o 0} f(x) = \infty$$, what does this imply about the graph of $$f(x)$$?

Accepted Answer

There is a vertical asymptote at $$x = 0.$$

Question 16

A population model is given by $$P(t) = \frac{1000}{1 + 0.01t}. $$What is $$\lim_{t 	o \infty} P(t)$$?

Accepted Answer

$$0$$

Question 17

Which of the following is a correct application of the limit laws?

Accepted Answer

$$\lim_{x 	o a} [f(x) + g(x)] = \lim_{x 	o a} f(x) + \lim_{x 	o a} g(x)$$, provided both limits exist.

Question 18

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

Accepted Answer

$$2$$

Question 19

Which of the following best describes a function with a removable discontinuity at $$x = a$$?

Accepted Answer

The limit $$\lim_{x 	o a} f(x)$$ exists, but $$f(a) is $$either undefined or not equal to the limit.

Limits and Continuity: Key Concepts and Theorems in Calculus

Study Guide - Practice Questions