Problem Set - 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

Limits

	Define the following and give an example for each: (a)   continuous at $p$, (b)   $\underset{x\to a}{\lim }f\left(x\right)$, (c)   continuous, (d)   uniformly continuous, (d)   Lipschiz continuous, (e)   derivative at $p$,
	For each of the following, guess the limit and then prove the guess by using the definition of limit: (a)   $\underset{x\to 4}{\lim }(\frac{1}{2}x-3)$, (b)   $\underset{x\to 0}{\lim }\frac{1}{1+x}$, (c)   $\underset{x\to 4}{\lim }\frac{1}{1+x^2}$, (d)   $\underset{x\to 1}{\lim }\frac{x^2-1}{x-1}$, (e)   $\underset{x\to 9}{\lim }\frac{x+1}{x^2+1}$, (f)   $\underset{x\to \infty }{\lim }\frac{\sin x}{x}$, (g)   $\underset{x\to 2}{\lim }\frac{2x^2+3x-8}{x^3-2x^2+x-12}$, (h)   $\underset{x\to \infty }{\lim }\frac{\log x+2x}{3x-5}$,
	Evaluate the following limits: (a)   $\underset{x\to 0}{\lim }x\cos \frac{1}{x^2}$, (b)   $\underset{x\to 0}{\lim }(\sqrt{5+x^2}-\sqrt{x^{-2}-1})$, (c)   $\underset{x\to 0}{\lim }\frac{\sqrt{1+x}-1}{x}$, (d)   $\underset{x\to \infty }{\lim }\frac{x^4+x}{x^4+1}$, (e)   $\underset{x\to \infty }{\lim }\frac{7x-1}{x^2}$, (f)   $\underset{x\to 0^{+}}{\lim }\frac{\sqrt{x}}{\sqrt{7+\sqrt{x+5}}}$, (g)   $\underset{x\to 1}{\lim }\frac{|x-1|+1}{x+|x+1|}$, (h)   $\underset{x\to \infty }{\lim }\frac{3x^2+1}{2x+1}$,
	Evaluate the following limits: (a)   $\underset{x\to 0}{\lim }{\displaystyle \frac{1-\cos x}{x+x^2}}$, (b)   $\underset{x\to \infty }{\lim }\frac{\log x}{x}$, (c)   $\underset{x\to 0+}{\lim }\sqrt{x}\hspace{0.17em}\log x$, (d)   $\underset{x\to 0+}{\lim }\frac{\sqrt{x}}{\log x}$, (e)   $\underset{x\to 0}{\lim }\frac{\sin x}{x}$, (f)   $\underset{x\to 0}{\lim }(\frac{1}{\arcsin x}-\frac{1}{\sin x})$.

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)