Free Practice Quiz Question List

2. Limits Online Quiz Questions

Use this free practice quiz with 20 questions to review 2. Limits, test your knowledge, and prepare for your next test or exam.

20 questions

True or false

1 point

True or false: A function can have a limit as x approaches a even when f(a) is undefined.

A
True
B
False

Choose one

1 point

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

A
0
B
4
C
2
D
Does not exist

Written response

1 point

Find the exact value of $\lim_{x\to0}\frac{\sqrt{x+1}-1}{x}$ .

Fill in the blank

1 point

A two-sided limit exists exactly when the left-hand and right-hand limits both exist and are .

Choose all

1 point

Select all situations that can cause a two-sided limit not to exist.

A
The left-hand and right-hand limits approach different values.
B
The function increases or decreases without bound near the input.
C
The function oscillates without approaching one value.
D
The function is undefined exactly at the input.

Choose one

1 point

What is the horizontal asymptote of $f(x)=\frac{5x^3-2x+1}{2x^3+x^2-4}$ ?

A
$y=0$
B
$y=2/5$
C
$y=5/2$
D
There is no horizontal asymptote

True or false

1 point

True or false: When $\lim_{x\to a}f(x)=+\infty$ , infinity is a real number that $f(x)$ reaches at $x=a$ .

A
True
B
False

Written response

1 point

What graph feature does the line x=4 represent for $f(x)=\frac{1}{(x-4)^2}$ ?

Choose all

1 point

When direct substitution gives $0/0$ , select all algebraic strategies that may be used to evaluate the limit.

A
Factoring and canceling a common factor
B
Rationalizing with a conjugate
C
Combining fractions algebraically
D
Treating $0/0$ as the value of the limit

Choose one

1 point

Let $f(x)=\begin{cases}0,&x<2\\3,&x\ge2\end{cases}$ . What is $\lim_{x\to2}f(x)$ ?

A
0
B
3
C
Does not exist
D
6

True or false

1 point

True or false: A function's graph may cross its horizontal asymptote.

A
True
B
False

Choose one

1 point

For $f(x)=\frac{x^3+1}{x^2+4}$ , what does the rational-function degree rule say about horizontal asymptotes?

A
$y=0$
B
No horizontal asymptote is determined by this rule
C
$y=1$
D
$y=3$

Open ended

1 point

Use the formal $\varepsilon$ - $\delta$ definition to prove that $\lim_{x\to2}(3x-1)=5$ . State an appropriate choice of $\delta$ in terms of $\varepsilon$ .

Choose one

1 point

What does the notation $\lim_{x\to a}f(x)$ describe?

A
The function must be evaluated only at $x=a$
B
The input approaches $a$ , without necessarily equaling $a$
C
The function must be defined at $x=a$
D
The function approaches infinity at $x=a$

Choose one

1 point

Which statement best describes the role of the value $f(a)$ when determining $\lim_{x\to a}f(x)$ ?

A
Always determines the limit
B
Determines the limit only when the function is continuous
C
Does not by itself determine the limit
D
Prevents the limit from existing

Choose one

1 point

Under what condition does the two-sided limit $\lim_{x\to a}f(x)$ exist?

A
Both one-sided limits exist and are equal
B
At least one one-sided limit is infinite
C
The function is defined at the approach point
D
The left-hand limit is larger than the right-hand limit

Choose one

1 point

Evaluate $\lim_{x\to3}\frac{x^2-9}{x-3}$ .

Written response

1 point

Find the exact value of $\lim_{x\to2}(3x^2-5x+4)$ . Enter the numerical value only; no rounding is needed.

Written response

1 point

Use the Squeeze Theorem to evaluate $\lim_{x\to 0}x\sin\left(\frac{1}{x}\right)$ . Enter the exact numerical value.

Fill in the blank

1 point

Suppose $\lim_{x\to a}f(x)=3$ and $\lim_{x\to a}g(x)=-2$ . Using the limit laws, determine $\lim_{x\to a}[2f(x)-g(x)^2]$ . Enter the exact value in .