$$\lim\limits_{x \to \infty} f(x) = L OR$$ $$\lim\limits_{x \to -\infty} f(x) = L$$

$$\lim\limits_{x \to \infty} \sin(x)$$

$$h(x) = \begin{cases} 2 & x \u003c 1 \\ 3 & x \geq 1 \end{cases}$$

$$\lim\limits_{x \to a} f(x) = f(a)$$

$$\lim\limits_{x \to a} f(x) = f(a)$$

Question 1

Which of the following is the correct definition of a horizontal asymptote for a function $$f(x)$$?

Accepted Answer

$$\lim\limits_{x 	o \infty} f(x) = L OR$$ $$\lim\limits_{x 	o -\infty} f(x) = L$$

Question 2

Evaluate $$\lim\limits_{x 	o \infty} \dfrac{3x^2$ + 7}{5x^2$ - x}.$$

Accepted Answer

$$\dfrac{3}{5}$$

Question 3

Which of the following functions has a removable discontinuity at $$x=1$$?

Accepted Answer

$$f(x) = \dfrac{x^2$ - 1}{x - 1}$$

Question 4

What is the value of $$\lim\limits_{x 	o \infty} \ln\left(\dfrac{2x+3}{5x+1}ight)$$?

Accepted Answer

$$\ln\left(\dfrac{2}{5}\right)$$

Question 5

Suppose $$f(x) is a $$rational function and $$\lim\limits_{x 	o a} f(x)$$ exists and is finite. What can you conclude about $$f(x) at$$ $$x=a$$?

Accepted Answer

$$f(x) is $$continuous at $$x=a if$$ $$f(a) is $$defined and equals the limit.

Question 6

Which of the following limits does NOT exist due to oscillation?

Accepted Answer

$$\lim\limits_{x 	o \infty} \sin(x)$$

Question 7

Which of the following is NOT a requirement for continuity of $$f(x) at$$ $$x=a$$?

Accepted Answer

$$f(x) is $$differentiable at $$x=a$$

Question 8

What type of discontinuity does the function $$f(x) = \dfrac{1}{x^3$}$$ have at $$x=0$$?

Accepted Answer

Infinite/Essential

Question 9

Given $$g(x) = \begin{cases} \dfrac{x^2$-4}{x-2} & x 
eq 2 \ 1 & x = 2 \end{cases}$$, what type of discontinuity does $$g(x)$$ have at $$x=2$$?

Accepted Answer

Removable

Question 10

Which of the following is an example of a function with a jump discontinuity at $$x=1$$?

Accepted Answer

$$h(x) = \begin{cases} 2 & x \u003c 1 \ 3 & x \geq 1 \end{cases}$$

Question 11

Evaluate $$\lim\limits_{x 	o \infty} x^2$ - x.$$

Accepted Answer

$$\infty$$

Question 12

Which of the following best describes the behavior of $$\lim\limits_{x 	o 0} \sin\left(\dfrac{1}{x}ight)$$?

Accepted Answer

The limit does not exist due to infinite oscillation.

Question 13

Suppose $$f(x) is $$continuous at $$x=a. $$Which of the following must be true?

Accepted Answer

$$\lim\limits_{x 	o a} f(x) = f(a)$$

Question 14

A function $$f(x) is $$defined as $$f(x) = \dfrac{1}{x}$$ for $$x 
eq 0. $$What type of discontinuity does $$f(x)$$ have at $$x=0$$?

Accepted Answer

Infinite/Essential

Question 15

Which of the following is true about the limit $$\lim\limits_{x 	o a} f(x)$$ for a trigonometric function $$f(x)$$ that is continuous at $$x=a$$?

Accepted Answer

$$\lim\limits_{x 	o a} f(x) = f(a)$$

Question 16

Which of the following is an example of a function that is continuous everywhere on its domain?

Accepted Answer

$$f(x) = \sin(x)$$

Infinite Limits, Horizontal Asymptotes, and Continuity in Calculus

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