Matematika SMP Oleh Mas Istanto

LIMIT TRIGONOMETRI

Selesaikanlah limit trigonometri berikut.

1.    $\lim_{x\rightarrow 0}\frac{\sin 6x-\sin 3x}{x}=....$

2. $\lim_{x\rightarrow 0}\frac{\1-\\cos x}{x}=....$

3. $\lim_{h\rightarrow 0}\frac{\sin \left ( x+h \right )-\sin x}{h}=....$

4.    $\lim_{x\rightarrow 0}\frac{\sin 9x}{\sin 3x}=....$

5.    $\lim_{x\rightarrow 0}\frac{x\sin x}{1-\cos x}=....$

Penyelesaian

1. $\lim_{x\rightarrow 0}\frac{\sin 6x-\sin 3x}{x}=\lim_{x\rightarrow 0}\left [ \frac{\sin 6x}{x} -\frac{\sin 3x}{x}\right ]$

$=\lim_{x\rightarrow 0}\frac{\sin 6x}{x}-\lim_{x\rightarrow 0}\frac{\sin 3x}{x}$

2. $\lim_{x\rightarrow 0}\frac{1-\cos x}{x}=\lim_{x\rightarrow 0}\left [ \frac{1-\cos x}{x}\cdot \frac{1+\cos x}{1+\cos x} \right ]$

$=\lim_{x\rightarrow 0}\frac{1-\cos ^{2}x}{x\left ( 1+\cos x \right )}$

$=\lim_{x\rightarrow 0}\frac{\sin ^{2}x}{x\left ( 1+\cos x \right )}$

$=\lim_{x\rightarrow 0}\left ( \frac{\sin x}{x}\cdot \frac{\sin x}{1+\cos x} \right )$

$=\lim_{x\rightarrow 0}\frac{\sin x}{x}\cdot \lim_{x\rightarrow 0}\frac{\sin x}{1+\cos x}$

$=1\cdot 0$

3. $\lim_{h\rightarrow 0}\frac{\sin \left ( x+h \right )-\sin x}{h}=\lim_{h\rightarrow 0}\frac{2\sin \frac{1}{2}\left ( x+h+x \right )\sin \frac{1}{2}\left ( x+h-h \right )}{h}$

$=\lim_{h\rightarrow 0}\frac{2\sin \frac{1}{2}\left ( 2x+h \right )\sin \frac{1}{2}h}{h}$

$=\lim_h\rightarrow 0}2\sin \frac{1}{2}\left ( 2x+h \right )\cdot \frac{\sin \frac{1}{2}h}{h}$

$=2\lim_{h\rightarrow 0}\sin \frac{1}{2}\left ( 2x+h \right )\cdot \lim_{h\rightarrow 0}\frac{\sin \frac{1}{2}h}{h}$

$=2\sin x\cdot \frac{1}{2}$

$=\sin x$

4. $\lim_{x\rightarrow 0}\frac{\sin 9x}{\sin 3x}=\lim_{x\rightarrow 0}\left ( \frac{\sin 9x}{9x}\cdot \frac{3x}{\sin 3x}\cdot \frac{9x}{3x} \right )$