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

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 $x\to a$

	Evaluate ${\displaystyle \underset{x\to 1}{lim}\left(6x^2-4x+3\right)}$.
	Evaluate ${\displaystyle \underset{x\to 7}{lim}\frac{x^2-49}{x-7}}$.
	Evaluate ${\displaystyle \underset{x\to 2}{lim}\frac{x^2-6x+8}{x-2}}$.
	Evaluate ${\displaystyle \underset{x\to -5}{lim}\frac{2x^2+9x-5}{x+5}}$.
	Evaluate ${\displaystyle \underset{x\to 1}{lim}\frac{x^3-1}{x-1}}$.
	Evaluate ${\displaystyle \underset{x\to 3}{lim}\frac{x^2-4x+3}{x^2-2x-3}}$.
	Evaluate ${\displaystyle \underset{x\to -2}{lim}\frac{x^3+8}{x+2}}$.
	Evaluate ${\displaystyle \underset{x\to 3}{lim}\frac{x^4-81}{x-3}}$.
	Evaluate ${\displaystyle \underset{x\to 5}{lim}\frac{x^5-3125}{x-5}}$.
	Evaluate ${\displaystyle \underset{x\to a}{lim}\frac{x^{12}-a^{12}}{x-a}}$.
	Evaluate ${\displaystyle \underset{x\to a}{lim}\frac{x^{5/2}-a^{5/2}}{x-a}}$.
	Evaluate ${\displaystyle \underset{x\to a}{lim}\frac{{\left(x+2\right)}^{5/3}-{\left(a+2\right)}^{5/3}}{x-a}}$.
	Evaluate ${\displaystyle \underset{x\to 4}{lim}\frac{x^3-64}{x^2-16}}$.
	Evaluate ${\displaystyle \underset{x\to 2}{lim}\frac{x^5-32}{x^3-8}}$.
	Evaluate ${\displaystyle \underset{x\to 1}{lim}\frac{x^n-1}{x-1}}$.
	Evaluate ${\displaystyle \underset{x\to a}{lim}\frac{\sqrt{x}-\sqrt{a}}{x-a}}$.
	Evaluate ${\displaystyle \underset{x\to 2}{lim}\frac{\sqrt{3-x}-1}{2-x}}$.
	Evaluate ${\displaystyle \underset{x\to a}{lim}\frac{\sqrt{a+2x}-\sqrt{3x}}{\sqrt{3a+x}-2\sqrt{x}}}$.
	Evaluate ${\displaystyle \underset{x\to a}{lim}\frac{x^n-a^n}{x-a}}$.

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)