Problem Set - Limits with Trigonometric Functions

## Problem Set - Limits with Trigonometric Functions

Last updates: 7 December 2009

## Limits with Trigonometric Functions

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