Question

Evaluate $\underset{x\to 0}{lim}\frac{\sin ax+bx}{ax+\sin bx}$.

Answer 1

We have,

$\underset{x\to 0}{lim}\frac{\sin ax+bx}{ax+\sin bx}$

$\underset{x\to 0}{lim}\frac{\frac{\sin ax}{ax}\times ax+bx}{ax+\frac{\sin bx}{bx}\times bx}$

$\underset{x\to 0}{lim}\frac{\frac{\sin ax}{ax}\times a+b}{a+\frac{\sin bx}{bx}\times b}$

We know that

${lim}_{x\to 0}\frac{\sin x}{x}=1$

Therefore,

$=\frac{1\times a+b}{a+1\times b}$

$=\frac{a+b}{a+b}=1$

Hence, this is the answer.