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Wize University Calculus 1 Textbook > Limits

Exam-Like Practice Questions

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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]?
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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}