# Math Help - limit-trigo

1. ## limit-trigo

calculate this limit ( no hospital ) :

$
\lim_{x\rightarrow \frac{\pi }{3}}(\frac{2cos(x)+1}{x-\frac{\pi }{3}})
$

2. Originally Posted by dhiab
calculate this limit ( no hospital ) :

$
\lim_{x\rightarrow \frac{\pi }{3}}(\frac{2cos(x)+1}{x-\frac{\pi }{3}})
$
Because:

$\displaystyle \lim_{x\to \frac{\pi }{3}}\left[2\cos(x)+1\right] \ne 0$

(since $\cos(\pi/3)=1/2$ ), The limit does not exist.

$
\displaystyle \lim_{x\to \frac{\pi }{3}}\left(\frac{2\cos(x)-1}{x-\frac{\pi }{3}}\right)
$

(Expanding the numerator as a Taylor series about $\pi/3$ this can be seen to be $-\sqrt{3}$ )

CB