Problem Set - Evaluating Limits when $x\to 0$

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: 7 December 2009

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 $f\left(x\right)={\left(mx+c\right)}^n$.

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)