limit as x->0 of (x*sin(x))/(1-cos(x))

[JoeTrumpet]

Can anyone help solve this analytically? I can do it graphically and with a table (and with a calculator), but I'm stumped analytically.

limit as x->0 of (x*sin(x))/(1-cos(x))

 

[pka]

\(\displaystyle \begin{array}{rcl}
\frac{{x\sin (x)}}{{1 - \cos (x)}} & = & \left( {\frac{{x\sin (x)}}{{1 - \cos (x)}}} \right)\left( {\frac{{1 + \cos (x)}}{{1 + \cos (x)}}} \right) \\
& = & \left( {\frac{{x\sin (x)\left( {1 + \cos (x)} \right)}}{{1 - \cos ^2 (x)}}} \right) \\
& = & \left( {\frac{{x\sin (x)\left( {1 + \cos (x)} \right)}}{{\sin ^2 (x)}}} \right) \\
& = & \left( {\frac{{x\left( {1 + \cos (x)} \right)}}{{\sin (x)}}} \right) \\
& = & \left( {\frac{x}{{\sin (x)}}} \right)\left( {1 + \cos (x)} \right)\quad \& \quad \lim _{x \to 0} \left( {\frac{x}{{\sin (x)}}} \right) = 1 \\
\end{array}\)

Now you finish.

 

[JoeTrumpet]

thanks a ton! I completely forgot that lim x->0 of x/sinx = 1 :wink: