# Using sine theta over theta equals 1 to find a limit

• January 23rd 2009, 09:49 AM
tom ato
Using sine theta over theta equals 1 to find a limit
$

\mathop {\lim }\limits_{x \to 2} {{{xcsc2x} \over {cos5x}}}

$

I keep running into trouble with the x in front of cosecant... I don't know what to do with it, so I always end up with a x/1 which yields zero! (The answer in the book says the limit is 1/2).
• January 23rd 2009, 09:53 AM
Jester
Quote:

Originally Posted by tom ato
$

\mathop {\lim }\limits_{x \to 2} {{{xcsc2x} \over {cos5x}}}

$

I keep running into trouble with the x in front of cosecant... I don't know what to do with it, so I always end up with a x/1 which yields zero! (The answer in the book says the limit is 1/2).

Re-write as $\lim_{x \to 2} \frac{x}{\sin 2x} \frac{1}{\cos 5 x }$

and use the fact that $\lim _{x \to 0} \frac{ \sin kx}{x} = k$