Question

Evaluate :$\underset{x\to 0}{lim}\frac{1-\cos x}{x}$

Answer 1

${lim}_{x\to 0}\frac{1-cosx}{x}={lim}_{x\to 0}\frac{2xsi{n}^{2}x/2}{x}$

$={lim}_{x\to 0}\frac{2si{n}^2\frac{x}{2}}{\left(\frac{x^2}{4}\right)}.\left(\frac{x}{4}\right)$

$={lim}_{x\to 0}2.\frac{x}{4}.(\frac{sin\frac{x}{2}}{\frac{x}{2}}{)}^2$

$=\left\{{lim}_{x\to 0}\frac{x}{2}\right\}.\left\{{lim}_{x\to 0}\right(\frac{sin\frac{x}{2}}{x/2}\left)\right\}.\left\{{lim}_{x\to 0}\right(\frac{sin\frac{x}{2}}{\frac{x}{2}}\left)\right\}$

$={0.1}^2$ $[\because {lim}_{x\to 0}\frac{sinx}{x}=1]$

=0