Problem Set: Limits
620-295 Semester I 2010

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 15 March 2010

(1) Evaluating Limits when $x\to 0$
(2) Evaluating Limits when $x\to a$
(3) Evaluating Limits when $x\to \infty$
(4) Limits with Exponential and Logarithm Functions
(5) Limits with Trigonometric Functions
(6) Limits with Inverse Trigonometric Functions
(7) Additional Limits

Evaluating Limits when $x\to 0$

	Evaluate ${\displaystyle \underset{x\to 0}{lim}{\left(x^2-2\right)}^2+6}$.
	Evaluate ${\displaystyle \underset{x\to 0}{lim}\frac{5x}{x}}$.
	Evaluate ${\displaystyle \underset{x\to 0}{lim}\frac{17x}{2x}}$.
	Evaluate ${\displaystyle \underset{x\to 0}{lim}\frac{-317x}{422x}}$.
	Evaluate ${\displaystyle \underset{x\to 0}{lim}\frac{-317x-3}{422x+5}}$.
	Evaluate ${\displaystyle \underset{h\to 0}{lim}\frac{\sqrt{x+h}-\sqrt{x}}{h}}$.
	Evaluate ${\displaystyle \underset{x\to 0}{lim}\frac{\sqrt{1+x+x^2}-1}{x}}$.
	Evaluate ${\displaystyle \underset{x\to 0}{lim}\frac{\sqrt{2+x}-\sqrt{2}}{x}}$.
	Evaluate ${\displaystyle \underset{h\to 0}{lim}\frac{1}{h}\left(\frac{1}{\sqrt{x+h}}-\frac{1}{\sqrt{x}}\right)}$.
	Evaluate ${\displaystyle \underset{x\to 0}{lim}\frac{2x}{\sqrt{a+x}-\sqrt{a-x}}}$.
	Evaluate ${\displaystyle \underset{x\to 0}{lim}\frac{\sqrt{1+x}-1}{x}}$.
	Evaluate ${\displaystyle \underset{x\to 0}{lim}\frac{x}{\sqrt{1+x}-1}}$.
	Evaluate ${\displaystyle \underset{x\to 0}{lim}\frac{e^x+e^{-x}-2}{x^2}}$.
	Evaluate ${\displaystyle \underset{\Delta x\to 0}{lim}\frac{f\left(x+\Delta x\right)-f\left(x\right)}{\left(x+\Delta x\right)-x}}$, when $f\left(x\right)=\sqrt{ax+b}$.
	Evaluate ${\displaystyle \underset{\Delta x\to 0}{lim}\frac{f\left(x+\Delta x\right)-f\left(x\right)}{\left(x+\Delta x\right)-x}}$, when $f\left(x\right)={\left(mx+c\right)}^n$.

Evaluating Limits when $x\to a$

	Evaluate ${\displaystyle \underset{x\to 1}{lim}\left(6x^2-4x+3\right)}$.
	Evaluate ${\displaystyle \underset{x\to 7}{lim}\frac{x^2-49}{x-7}}$.
	Evaluate ${\displaystyle \underset{x\to 2}{lim}\frac{x^2-6x+8}{x-2}}$.
	Evaluate ${\displaystyle \underset{x\to -5}{lim}\frac{2x^2+9x-5}{x+5}}$.
	Evaluate ${\displaystyle \underset{x\to 1}{lim}\frac{x^3-1}{x-1}}$.
	Evaluate ${\displaystyle \underset{x\to 3}{lim}\frac{x^2-4x+3}{x^2-2x-3}}$.
	Evaluate ${\displaystyle \underset{x\to -2}{lim}\frac{x^3+8}{x+2}}$.
	Evaluate ${\displaystyle \underset{x\to 3}{lim}\frac{x^4-81}{x-3}}$.
	Evaluate ${\displaystyle \underset{x\to 5}{lim}\frac{x^5-3125}{x-5}}$.
	Evaluate ${\displaystyle \underset{x\to a}{lim}\frac{x^{12}-a^{12}}{x-a}}$.
	Evaluate ${\displaystyle \underset{x\to a}{lim}\frac{x^{5/2}-a^{5/2}}{x-a}}$.
	Evaluate ${\displaystyle \underset{x\to a}{lim}\frac{{\left(x+2\right)}^{5/3}-{\left(a+2\right)}^{5/3}}{x-a}}$.
	Evaluate ${\displaystyle \underset{x\to 4}{lim}\frac{x^3-64}{x^2-16}}$.
	Evaluate ${\displaystyle \underset{x\to 2}{lim}\frac{x^5-32}{x^3-8}}$.
	Evaluate ${\displaystyle \underset{x\to 1}{lim}\frac{x^n-1}{x-1}}$.
	Evaluate ${\displaystyle \underset{x\to a}{lim}\frac{\sqrt{x}-\sqrt{a}}{x-a}}$.
	Evaluate ${\displaystyle \underset{x\to 2}{lim}\frac{\sqrt{3-x}-1}{2-x}}$.
	Evaluate ${\displaystyle \underset{x\to a}{lim}\frac{\sqrt{a+2x}-\sqrt{3x}}{\sqrt{3a+x}-2\sqrt{x}}}$.
	Evaluate ${\displaystyle \underset{x\to a}{lim}\frac{x^n-a^n}{x-a}}$.

Evaluating Limits when $x\to \infty$

	Evaluate ${\displaystyle \underset{x\to \infty }{lim}\frac{x+2}{x-2}}$.
	Evaluate ${\displaystyle \underset{x\to \infty }{lim}\frac{3x^2+2x-5}{5x^2+3x+1}}$.
	Evaluate ${\displaystyle \underset{x\to \infty }{lim}\frac{x^2-7x+11}{3x^2+10}}$.
	Evaluate ${\displaystyle \underset{x\to \infty }{lim}\frac{2x^3-5x+7}{7x^3+x^2-6}}$.
	Evaluate ${\displaystyle \underset{x\to \infty }{lim}\frac{2x^3-5x+7}{7x^3+x^2-6}}$.
	Evaluate ${\displaystyle \underset{x\to \infty }{lim}\frac{\left(3x-1\right)\left(4x-5\right)}{\left(x-6\right)\left(x-3\right)}}$.
	Evaluate ${\displaystyle \underset{n\to \infty }{lim}\left(\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+\dots +\frac{1}{3^n}\right)}$.
	Evaluate ${\displaystyle \underset{x\to \infty }{lim}\frac{x}{\sqrt{4x^2-1}-1}}$.
	Evaluate ${\displaystyle \underset{x\to -\infty }{lim}{2}^x}$.
	Evaluate ${\displaystyle \underset{n\to \infty }{lim}{\left(1+\frac{1}{n}\right)}^n}$.
	Evaluate ${\displaystyle \underset{t\to \infty }{lim}\frac{t+1}{t^2+1}}$.
	Evaluate ${\displaystyle \underset{n\to \infty }{lim}\sqrt{n^2+1}+n}$.
	Evaluate ${\displaystyle \underset{n\to \infty }{lim}\sqrt{n^2+n}+n}$.

Limits with Exponential and Logarithm Functions

	Evaluate ${\displaystyle \underset{x\to 0}{lim}\frac{e^x-1}{x}}$.
	Evaluate ${\displaystyle \underset{x\to 0}{lim}\frac{a^x-1}{x}}$.
	Evaluate ${\displaystyle \underset{x\to 0}{lim}\frac{\ln \left(1+x\right)}{x}}$.
	Evaluate ${\displaystyle \underset{x\to 0}{lim}{\left(1+x\right)}^{1/x}}$.
	Evaluate ${\displaystyle \underset{x\to 0}{lim}\frac{a^x-b^x}{x}}$.
	Evaluate ${\displaystyle \underset{x\to 0}{lim}\frac{e^x+e^{-\mathrm{x}}-2}{x^2}}$.
	Evaluate ${\displaystyle \underset{x\to -\infty }{lim}{2}^x}$.
	Explain why ${\displaystyle \underset{x\to -1}{lim}\ln \left(x\right)}$ does not exist.
	Explain why ${\displaystyle \underset{x\to 0}{lim}{2}^{1/x}}$ does not exist.
	Explain why ${\displaystyle \underset{x\to 1}{lim}{2}^{1/\left(x-1\right)}}$ does not exist.
	Evaluate ${\displaystyle \underset{\Delta x\to 0}{lim}\frac{f\left(x+\Delta x\right)-f\left(x\right)}{\left(x+\Delta x\right)-x}}$ where ${\displaystyle f\left(x\right)=e^{\sqrt{x}}}$.
	Evaluate ${\displaystyle \underset{\Delta x\to 0}{lim}\frac{f\left(x+\Delta x\right)-f\left(x\right)}{\left(x+\Delta x\right)-x}}$ where ${\displaystyle f\left(x\right)=ln\left(ax+b\right)}$.
	Evaluate ${\displaystyle \underset{\Delta x\to 0}{lim}\frac{f\left(x+\Delta x\right)-f\left(x\right)}{\left(x+\Delta x\right)-x}}$ where ${\displaystyle f\left(x\right)=x^x}$.

Limits with Trigonometric Functions

	Evaluate ${\displaystyle \underset{x\to 0}{lim}\frac{\sin 3x}{4x}}$.
	Evaluate ${\displaystyle \underset{x\to 0}{lim}\frac{\sin x\cos x}{3x}}$.
	Evaluate ${\displaystyle \underset{x\to 0}{lim}\frac{\tan x}{x}}$.
	Evaluate ${\displaystyle \underset{x\to 0}{lim}\frac{1-\cos x}{{\sin }^{2}x}}$.
	Evaluate ${\displaystyle \underset{x\to 0}{lim}\frac{\tan ax}{\tan bx}}$.
	Evaluate ${\displaystyle \underset{x\to 0}{lim}\frac{\sin \left(x/4\right)}{x}}$.
	Evaluate ${\displaystyle \underset{x\to 0}{lim}\frac{\sin mx}{\tan nx}}$.
	Evaluate ${\displaystyle \underset{\theta \to 0}{lim}\frac{1-\cos 6\theta }{\theta }}$.
	Evaluate ${\displaystyle \underset{x\to 0}{lim}\frac{1-\cos 2x}{3{\tan }^{2}x}}$.
	Evaluate ${\displaystyle \underset{x\to 0}{lim}\frac{{\cos }^{2}x}{1-\sin x}}$.
	Evaluate ${\displaystyle \underset{x\to 0}{lim}\frac{\tan 2x-x}{3x-\sin x}}$.
	Evaluate ${\displaystyle \underset{x\to a}{lim}\frac{\sin x-\sin a}{x-a}}$.
	Evaluate ${\displaystyle \underset{x\to 0}{lim}\frac{\sin 5x-\sin 3x}{\sin x}}$.
	Evaluate ${\displaystyle \underset{x\to 0}{lim}\frac{\tan 3x-2x}{3x-{\sin }^{2}x}}$.
	Evaluate ${\displaystyle \underset{x\to 0}{lim}\frac{x^2-\tan 2x}{\tan x}}$.
	Evaluate ${\displaystyle \underset{x\to \pi /4}{lim}\frac{1-\tan x}{x-\pi /4}}$.
	Evaluate ${\displaystyle \underset{x\to 0}{lim}\frac{\tan \left(x/2\right)}{3x}}$.
	Evaluate ${\displaystyle \underset{x\to 0}{lim}\frac{1-\cos 2x+{\tan }^{2}x}{x\sin x}}$.
	Show that if ${\displaystyle \underset{x\to 0}{lim}kx\csc x=\underset{x\to 0}{lim}x\csc kx}$, then k = ± 1.
	Evaluate ${\displaystyle \underset{h\to 0}{lim}\frac{\sin \left(a+h\right)-\sin a}{h}}$.
	Evaluate ${\displaystyle \underset{h\to \mathrm{\infty }}{lim}\frac{\cos \left(\pi /h\right)}{h-2}}$.

Limits with Inverse Trigonometric Functions

	Evaluate ${\displaystyle \underset{x\to 1}{lim}\frac{1-x}{{\arccos }^{2}x}}$.
	Evaluate ${\displaystyle \underset{x\to 1/\sqrt{2}}{lim}\frac{x-\cos \left(\arcsin x\right)}{1-\tan \left(\arcsin x\right)}}$.
	Evaluate ${\displaystyle \underset{x\to 0}{lim}\frac{x\left(1-\sqrt{1-x^2}\right)}{{\arcsin }^3\left(x\right)\sqrt{1-x^2}}}$.
	Evaluate ${\displaystyle \underset{x\to 1}{lim}\frac{1-x}{\pi -2\arcsin x}}$.
	Evaluate ${\displaystyle \underset{x\to 1}{lim}\frac{\arctan 2x}{\sin 3x}}$.

Additional Limits

	Let $p\in {ℝ}_{>0}$. Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\frac{1}{n^p}}$.
	Let $r\in ℝ$ such that $\left|r\right|<1$. Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }{r}^n}$.
	Let $a\in {ℝ}_{>0}$. Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }{a}^{\frac{1}{n}}}$.
	Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }{n}^{\frac{1}{n}}}$.
	Let $a\in {ℝ}_{>0}$. Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\frac{a^n}{n\text{!}}}$.
	Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\frac{\log n}{n^p}}$.
	Let $a\in ℝ$. Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }{(1+\frac{a}{n})}^n}$.
	Let $a,p\in ℝ$ with $a>1$ and $p>0$. Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\frac{n^p}{a^n}}$.
	Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\frac{n}{2n+3}}$.
	Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\frac{n}{n+1}-\frac{n+1}{n}}$.
	Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\frac{1-n}{n^3}}$.
	Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\frac{3n-1}{2n+5}}$.
	Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\frac{n+1}{\sqrt{n}}}$.
	Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\frac{\sqrt{n}}{n+1}}$.
	Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }1+{(-1)}^{n+1}}$.
	Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }{n}^{{(-1)}^n}}$.
	Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }{a}_n}$ when $a_n=\sqrt{3}$.
	Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\frac{n}{2^n}}$.
	Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\cos \frac{n\pi }{2}}$.
	Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }(1+{(-1)}^n)\frac{1}{n}}$.
	Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\frac{4n^2-2n+\cos n}{3n^2+7n+6}}$.
	Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\frac{3n-5}{\sqrt{2n^2+1}+\sqrt{3n^2+4}}}$.
	Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\frac{2n^2+6n+2}{3n^3-n^2-n}}$.
	Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\sqrt{n^2+3n}-\sqrt{n^2+4}}$.
	Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\sqrt[n]{2n}}$.
	Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }{(1-\frac{1}{n^2})}^n}$.
	Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\sqrt[n]{n^2}}$.
	Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\frac{\log n+1}{n}}$.
	Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\frac{\log (n+1)}{n}}$.
	Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\frac{\log n}{\sqrt{n}}}$.
	Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\frac{n^n}{{(n+3)}^{n+1}}}$.
	Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\frac{{\left(2n\right)}^3}{3^n}}$.
	Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }{\left(\frac{n}{n+3}\right)}^n}$.
	Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\frac{n^5}{2^n}}$.
	Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\frac{2^n}{\left(2n\right)\text{!}}}$.
	Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\frac{n\text{!}}{{10}^{6n}}}$.
	Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\sqrt[n]{2n+1}}$.
	Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\sqrt[n]{n^2+n}}$.
	Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\frac{n\text{!}}{n^n}}$.
	Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\sin (\frac{\pi }{3}+\frac{1}{n})}$.
	Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\tanh n}$.
	Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }n\sin \left(\frac{\pi }{n}\right)}$.
	Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }(n+1)\log \left(\frac{n-1}{n}\right)}$.
	Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\arctan \left(\cos \left(\frac{\pi }{n}\right)\right)}$.
	Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\frac{1}{n}\cosh \left(\log \left(\frac{2n^2+1}{n}\right)\right)}$.
	Evaluate ${\displaystyle \underset{x\to 0}{\lim }\frac{1-\frac{1}{2}{x}^2-\cos x}{x^4}}$.
	Evaluate ${\displaystyle \underset{x\to 0}{\lim }\frac{x-\arctan x}{x^3}}$.
	Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\frac{3^n-n\text{!}}{n\text{!}+4^n-2}}$.
	Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\frac{\cos n+\log n+n^5}{5n^5-2}}$.
	Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\frac{{(-1)}^n{n}^{\frac{1}{n}}}{3^{\frac{1}{n}}}}$.
	Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }n(\frac{\pi }{2}-\arctan n)}$.
	Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\frac{2n^3-5^n+2^n}{5^n-5}}$.
	Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\frac{n\text{!}+3^n}{2^n+3}}$.
	Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\sqrt[n]{4+n}}$.
	Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\frac{\log (n^2+3)}{\log (3n+5)}}$.
	Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\log (n^2+1)-\log (n^2+3)}$.
	Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\frac{n\text{!}+3^n}{3^n+\log n}}$.
	Evaluate ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }{\left(\frac{n-1}{n-1}\right)}^n}$.
	Let $p\in {ℝ}_{>0}$. Evaluate ${\displaystyle \underset{x\to \mathrm{\infty }}{\lim }{x}^p{e}^{-x}}$.

References

[Ca] S. Carnie, 620-143 Applied Mathematics, Course materials, 2006 and 2007.

[Ho] C. Hodgson, 620-194 Mathematics B and 620-211 Mathematics 2 Notes, Semester 1, 2005.