Ask Me Help Desk - Limits/Trig Identities

Ask Me Help Desk (https://www.askmehelpdesk.com/forum.php)

- Math & Sciences (https://www.askmehelpdesk.com/forumdisplay.php?f=402)

- - Limits/Trig Identities (https://www.askmehelpdesk.com/showthread.php?t=110257)

Limits/Trig Identities

I need to find the limit as x approaches 0 for tan 6x / sin 2x

Try L'Hopital's Rule.

Here's a way without L'Hopital.

$\lim_{x\to\0}\frac{tan(6x)}{sin(2x)}$

$\lim_{x\to\0}\frac{sin(6x)}{cos(6x)}\cdot\frac{1}{ sin(2x)}$

$\lim_{x\to\0}\frac{sin(6x)}{sin(2x)}\cdot\frac{1}{ cos(6x)}$

$\lim_{x\to\0}\frac{\frac{sin(6x)}{x}}{\frac{sin(2x )}{x}}\cdot\lim_{x\to\0}\frac{1}{cos(6x)}$

Since $\lim_{x\to\0}\frac{sin(x)}{x}=1$ :

$\lim_{x\to\0}\frac{6\cdot\frac{sin(6x)}{6x}}{2\tim \frac{sin(2x)}{2x}}\cdot\lim_{x\to\0}\frac{1}{cos( 6x)}=\frac{6\cdot{1}}{5\cdot{1}}\cdot{1}=\fbox{3}$

All times are GMT -7. The time now is 02:51 PM.