Question

${lim}_{x\to 0}\frac{1-cosmx}{x^2}$

Answer 1

${lim}_{x\to 0}\frac{1-cosmx}{x^2}$

$={lim}_{x\to 0}\frac{2si{n}^2\frac{mx}{2}}{x^2}=2{\left({lim}_{x\to 0}\frac{sin\frac{mx}{2}}{x}\right)}^2=2{\left({lim}_{x\to 0}\frac{sin\frac{mx}{2}}{\frac{mx}{2}}\right)}^2\times {\left(\frac{m}{2}\right)}^2$

$=2\times \frac{m^2}{4}=\frac{m^2}{2}[\because {lim}_{x\to 0}\frac{sinx}{x}=1]$

$=\frac{m^2}{2}$