Answer
$\displaystyle\lim_{\theta\rightarrow 0} \dfrac{\cos\theta-1}{\theta}=0$
Work Step by Step
We have to estimate the limit:
$\displaystyle\lim_{\theta\rightarrow 0} \dfrac{\cos\theta-1}{\theta}$
Compute $\dfrac{\cos\theta-1}{\theta}$ for values of $\theta$ close to 0:
$\dfrac{\cos (-0.1)-1}{-0.1}\approx 0.0499583$
$\dfrac{\cos (-0.01)-1}{-0.01}\approx 0.0049996$
$\dfrac{\cos (-0.001)-1}{-0.001}\approx 0.0005$
$\dfrac{\cos (-0.0001)-1}{0.0001}\approx 0.00005$
$\dfrac{\cos (0.0001)-1}{0.0001}\approx 0.00005$
$\dfrac{\cos (0.001)-1}{0.001}\approx 0.0005$
$\dfrac{\cos (0.01)-1}{0.01}\approx 0.0049996$
$\dfrac{\cos (0.1)-1}{0.1}\approx 0.0499583$
Therefore we have:
$\displaystyle\lim_{\theta\rightarrow 0} \dfrac{\cos\theta-1}{\theta}=0$
