AI TutorHigh School
SAT
ACT
University
MCAT
LSAT
DAT
Wize University Calculus 1 Textbook > Limits

Exam-Like Practice Questions

The Epsilon Delta Definition of a LimitPrevious Section
Derivatives as a Rate of ChangeNext Section
Popular Courses
Find My Course
Evaluate the limit lim⁡x→1x2−1∣x−1∣\displaystyle\lim_{x\to1}\frac{x^2-1}{\left|x-1\right|}
State the asymptotes of the function f(x)=3x2−54x2−1f\left(x\right)=\frac{3x^2-5}{4x^2-1}.
Find the value of lim⁡x→0sin⁡2xsin⁡3x\displaystyle \lim_{x\rightarrow0}\frac{\sin2x}{\sin3x}.
Find the following limit:

lim⁡x→2−5(x+1)x−2\displaystyle\lim_{x\rightarrow 2}\frac{-5(x+1)}{x-2}

lim⁡x→−∞2−x−2x23x2+5\displaystyle\lim_{x\rightarrow -\infty}\dfrac{2-x-2x^2}{3x^2+5}

lim⁡x→∞x2+2x2−3x+1\displaystyle\lim_{x\to\infty}\dfrac{x^2+2}{x^2-3x+1}

lim⁡x→1x5−1x−1\displaystyle\lim_{x\rightarrow1}\dfrac{x^5-1}{x-1}

Evaluate the left and right hand limits.
a)
lim⁡x→7−5∣x−7∣x−7\displaystyle \lim_{x \rightarrow 7^-} \frac{5|x-7|}{x-7}

b)
lim⁡x→7+5∣x−7∣x−7\displaystyle \lim_{x \rightarrow 7^+} \frac{5|x-7|}{x-7}

c) Does the two-sided limit exist?

lim⁡x→75∣x−7∣x−7\displaystyle \lim_{x \rightarrow 7} \frac{5|x-7|}{x-7}
Evaluate the following limits.

lim⁡x→2x+7−3x−2\displaystyle \lim_{x\rightarrow 2}\frac{\sqrt{x+7}-3}{x-2}

Evaluate the limit lim⁡x→0 ∣4x−1∣−∣4x+1∣x\displaystyle \lim_{x\to0}\ \frac{\left|4x-1\right|-\left|4x+1\right|}{x}.

Practice: Evaluating Limits

Given the following function, evaluate the limits below.
f(x)={−3x+5if  x<−1−10if  x=−19−x2if  −1<x≤21−∣2x−3∣if  x>2f\left(x\right)= \begin{cases} -3x+5 & \text{if } \ x<-1 \\ -10 & \text{if }\ x=-1\\ 9-x^2 & \text{if }\ -1< x\le2\\ 1-|2x-3| & \text{if } \ x>2 \end{cases}

Enter DNE if the limit does not exist.
Does 2x3−2+cos⁡(x)=02x^3-2+\cos(x)=0 have a root in (0,1) ?


Does x3−1+sin⁡(x)=0x^3-1+\sin\left(x\right)=0 have a root in [0,1]\left[0,1\right]?
checklist
Mark Yourself Question
  1. Grab a piece of paper and try this problem yourself.
  2. When you're done, check the "I have answered this question" box below.
  3. View the solution and report whether you got it right or wrong.
Show that there is at least one solution to the equation ln⁡x=esin⁡(πx2)\ln x=e^{\sin\left(\frac{\pi x}{2}\right)}.

Evaluate the following limit:
lim⁡x→0sin⁡(3x)cos⁡(2x)sin⁡(5x)\lim_{x\rightarrow0}\frac{\sin(3x)\cos(2x)}{\sin(5x)}




Determine lim⁡x→0x2tan⁡2 2x\lim\limits_{x\rightarrow0}\frac{x^2}{\tan^2\ 2x}.
Suppose that f(x) is a function such that

3x2−x+6≤f(x)≤Ax−33x^2-x+6 \leq f(x) \leq Ax-3
for x close to 4. In order for us to use the Squeeze Theorem to find lim⁡x→4f(x)\lim\limits_{x\to4}f(x), what is the value of A and the limit lim⁡x→4f(x)\lim\limits_{x\to4}f(x)?


Evaluate lim⁡x→∞ e1x⋅cos⁡(x2−2)ln⁡x\displaystyle\lim_{x\to\infty}\ \frac{e^{\frac{1}{x}}\cdot\cos\left(x^2-2\right)}{\ln x}.
Find lim⁡x→0(xsin⁡1x2+x)\displaystyle \lim_{x\rightarrow 0}\left(x\sin{\frac{1}{x^2+x}}\right)
For what values of aa is the following function continuous on (−∞,∞)(-\infty,\infty).

f(x)={ax3−x2+sin⁡(πx)x<2x2−(2+a)x+2ax−1x≥2f(x)= \begin{cases} ax^3-x^2+\sin(\pi x) & x<2 \\ \frac{x^2-(2+a)x+2a}{x-1} & x\geq 2 \end{cases}