Problem Set - Limits

Problem Set - Limits

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia


Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA

Last updates: 7 December 2009


Define the following and give an example for each:
(a)   continuous at p ,
(b)   limx af(x) ,
(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)   limx4 (12x-3) ,
(b)   limx0 11+x ,
(c)   limx4 11+x2 ,
(d)   limx1 x2-1 x-1 ,
(e)   limx9 x+1 x2+1 ,
(f)   limx sinxx ,
(g)   limx2 2x2+3x-8 x3 -2x2+x-12 ,
(h)   limx logx+2x 3x-5 ,

Evaluate the following limits:
(a)   limx0 xcos1x2 ,
(b)   limx0 ( 5+x2 - x-2-1 ) ,
(c)   limx0 1+x-1 x ,
(d)   limx x4+x x4+1 ,
(e)   limx 7x-1 x2 ,
(f)   limx0+ x 7+x+5 ,
(g)   limx1 |x-1|+1 x+|x+1| ,
(h)   limx 3x2+1 2x+1 ,

Evaluate the following limits:
(a)   limx0 1-cosx x+x2 ,
(b)   limx logx x ,
(c)   limx0+ x logx ,
(d)   limx0+ x logx ,
(e)   limx0 sinx x ,
(f)   limx0 ( 1 arcsinx - 1 sinx ) .

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)