Question

${lim}_{x\to o\left(\frac{sinax}{sinbx}\right)}a,b\pm 0.$

Answer 1

We have,

${lim}_{x\to 0}\left(\frac{\sin ax}{\sin bx}\right)$

Then,

${lim}_{x\to 0}\left(\frac{\frac{ax\sin ax}{ax}}{\frac{bx\sin bx}{bx}}\right)$

$\Rightarrow \frac{ax}{bx}{lim}_{x\to 0}\left(\frac{\frac{\sin ax}{ax}}{\frac{\sin bx}{bx}}\right)$

$\Rightarrow \frac{ax}{bx}\left(\frac{{lim}_{x\to 0}\frac{\sin ax}{ax}}{{lim}_{x\to 0}\frac{\sin bx}{bx}}\right)({lim}_{x\to \infty }\frac{\mathrm{sinx}}{x}=1)$

$\Rightarrow \frac{a}{b}\times \frac{1}{1}$

$\Rightarrow \frac{a}{b}$

Hence, tis is the answer.