Problem Set - Limits with Trigonometric 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 Trigonometric Functions

	Evaluate ${\displaystyle \underset{x\to 0}{lim}\frac{\sin 3x}{4x}}$.
	Evaluate ${\displaystyle \underset{x\to 0}{lim}\frac{\sin x\cos x}{3x}}$.
	Evaluate ${\displaystyle \underset{x\to 0}{lim}\frac{\tan x}{x}}$.
	Evaluate ${\displaystyle \underset{x\to 0}{lim}\frac{1-\cos x}{{\sin }^{2}x}}$.
	Evaluate ${\displaystyle \underset{x\to 0}{lim}\frac{\tan ax}{\tan bx}}$.
	Evaluate ${\displaystyle \underset{x\to 0}{lim}\frac{\sin \left(x/4\right)}{x}}$.
	Evaluate ${\displaystyle \underset{x\to 0}{lim}\frac{\sin mx}{\tan nx}}$.
	Evaluate ${\displaystyle \underset{\theta \to 0}{lim}\frac{1-\cos 6\theta }{\theta }}$.
	Evaluate ${\displaystyle \underset{x\to 0}{lim}\frac{1-\cos 2x}{3{\tan }^{2}x}}$.
	Evaluate ${\displaystyle \underset{x\to 0}{lim}\frac{{\cos }^{2}x}{1-\sin x}}$.
	Evaluate ${\displaystyle \underset{x\to 0}{lim}\frac{\tan 2x-x}{3x-\sin x}}$.
	Evaluate ${\displaystyle \underset{x\to a}{lim}\frac{\sin x-\sin a}{x-a}}$.
	Evaluate ${\displaystyle \underset{x\to 0}{lim}\frac{\sin 5x-\sin 3x}{\sin x}}$.
	Evaluate ${\displaystyle \underset{x\to 0}{lim}\frac{\tan 3x-2x}{3x-{\sin }^{2}x}}$.
	Evaluate ${\displaystyle \underset{x\to 0}{lim}\frac{x^2-\tan 2x}{\tan x}}$.
	Evaluate ${\displaystyle \underset{x\to \pi /4}{lim}\frac{1-\tan x}{x-\pi /4}}$.
	Evaluate ${\displaystyle \underset{x\to 0}{lim}\frac{\tan \left(x/2\right)}{3x}}$.
	Evaluate ${\displaystyle \underset{x\to 0}{lim}\frac{1-\cos 2x+{\tan }^{2}x}{x\sin x}}$.
	Show that if ${\displaystyle \underset{x\to 0}{lim}kx\csc x=\underset{x\to 0}{lim}x\csc kx}$, then k = ± 1.
	Evaluate ${\displaystyle \underset{h\to 0}{lim}\frac{\sin \left(a+h\right)-\sin a}{h}}$.
	Evaluate ${\displaystyle \underset{h\to \mathrm{\infty }}{lim}\frac{\cos \left(\pi /h\right)}{h-2}}$.

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)