Problem Set - Existence of Limits

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

Existence of Limits

	Explain why ${\displaystyle \underset{x\to 0}{\lim }\left(\frac{1}{x}\right)}$ does not exist.
	Explain why ${\displaystyle \underset{x\to \pi /2}{\lim }\tan \left(x\right)}$ does not exist.
	Explain why ${\displaystyle \underset{x\to \pi /2}{\lim }\sec \left(x\right)}$ does not exist.
	Explain why ${\displaystyle \underset{x\to 0}{\lim }\csc \left(x\right)}$ does not exist.
	Explain why ${\displaystyle \underset{x\to -1}{\lim }\ln \left(x\right)}$ does not exist.
	Explain why ${\displaystyle \underset{x\to 0}{\lim }\sin \left(\frac{1}{x}\right)}$ does not exist.
	Explain why ${\displaystyle \underset{x\to \mathrm{\infty }}{\lim }\cos \left(x\right)}$ does not exist.
	Explain why ${\displaystyle \underset{x\to 0}{\lim }\mathrm{sgn}\left(x\right)}$ does not exist, where $\mathrm{sgn}\left(x\right)=\left\{\begin{array}{ll}1\text{,}& \text{if }x>0\\ 0\text{,}& \text{if }x=0\\ -1\text{,}& \text{if }x<0\end{array}\right.$.
	Explain why ${\displaystyle \underset{x\to 0}{\lim }{2}^{1/x}}$ does not exist.
	Explain why ${\displaystyle \underset{x\to 1}{\lim }{2}^{1/\left(1-x\right)}}$ does not exist.

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)