Problem Set - Limits with Exponential and Logarithm Functions

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

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 ${\displaystyle f\left(x\right)=x^x}$.

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)