Question

evaluate the limit lim x—->0 sin(2x)/x^2+x

Answer 1

Recall that

$\displaystyle\lim_{x\to0}\frac{\sin ax}{ax}=1$

for $a\neq0$.

We have

$\displaystyle\lim_{x\to0}\frac{\sin2x}{x^2+x}=\lim_{x\to0}\frac{\sin2x}{2x}\cdot\lim_{x\to0}\frac2{x+1}=2$