Question

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

Answer 1

We have,

${lim}_{x\to 0}\frac{1-\cos mx}{1-\cos nx}$

$\Rightarrow {lim}_{x\to 0}\frac{1-1+2{\sin }^2\frac{mx}{2}}{1-1+2si{n}^2\frac{nx}{2}}$

$\Rightarrow {lim}_{x\to 0}\frac{{\sin }^2\frac{mx}{2}}{si{n}^2\frac{nx}{2}}$

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

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

$\Rightarrow \frac{1}{1}\times \frac{m^2}{n^2}$

$\Rightarrow \frac{m^2}{n^2}$

Hence, this is the answer.