Question

Evaluate the limit:

$\underset{x\to 0}{lim}\frac{5x+4\sin 3x}{4\sin 2x+7x}$

Answer 1

We have,

$\underset{x\to 0}{lim}\frac{5x+4\sin 3x}{4\sin 2x+7x}$

It becomes 0/0 form

Dividing numerator and denominator by $x$

$=\underset{x\to 0}{lim}\frac{\frac{5x}{x}+\frac{4\sin 3x}{x}}{\frac{4\sin 2x}{x}+\frac{7x}{x}}$

$=\underset{x\to 0}{lim}\frac{5+12\times \frac{\sin 3x}{3x}}{8\times \frac{\sin 2x}{2x}+7}$

$=\frac{5+12}{8+7}[\because \underset{x\to 0}{lim}\frac{\sin x}{x}=1]$

$=\frac{17}{15}$