# Limit of [1 - cos (x)]/x as x --> 0

• Jun 30th 2011, 04:45 AM
khamaar
Limit of [1 - cos (x)]/x as x --> 0
WHAT IS LIMIT OF , [1 - cos (x)]/x such that "x approaches zero"

i think the answer is "limit doesnt exist"..

The book says its 0.
• Jun 30th 2011, 04:55 AM
e^(i*pi)
Re: Need help with solving a limit....
Are you allowed to use L'Hopital's rule?
• Jun 30th 2011, 04:59 AM
Prove It
Re: Need help with solving a limit....
Quote:

Originally Posted by khamaar
WHAT IS LIMIT OF , [1 - cos (x)]/x such that "x approaches zero"

i think the answer is "limit doesnt exist"..

The book says its 0.

\displaystyle \begin{align*} \lim_{x \to 0}\frac{1 - \cos{x}}{x} &= \lim_{x \to 0}\frac{(1 - \cos{x})(1 + \cos{x})}{x(1 + \cos{x})} \\ &= \lim_{x \to 0}\frac{1 - \cos^2{x}}{x(1 + \cos{x})} \\ &= \lim_{x \to 0}\frac{\sin^2{x}}{x(1 + \cos{x})} \\ &= \lim_{x \to 0}\frac{\sin{x}}{x}\cdot\lim_{x \to 0}\frac{\sin{x}}{1 + \cos{x}} \\ &= 1\cdot 0 \\ &= 0 \end{align*}
• Jun 30th 2011, 07:06 AM
khamaar
Re: Need help with solving a limit....
thank you very much, case closed.

Thanks again...Hugs!!