$\lim_{x \rightarrow 0} \dfrac{x-\sin x}{x+\sin x}$

Question

I did this using L'Hospital's rule but I'm trying to figure out a way to do it without L'Hospital's just using the fact that $\lim_{x \rightarrow 0} \frac{\sin x}{x}=1$

Answer 1

You can just divide the numerator and denominator by $x$ to get

$$\lim\limits_{x\rightarrow 0}\,\frac{1-\frac{\sin x}{x}}{1+\frac{\sin x}{x}}$$

Answer 2

We have

$sin(x)=x+x\epsilon(x)$ with

$\lim_{x\to 0}\epsilon(x)=0$

thus the function becomes

$$\frac{-x\epsilon(x)}{2x+x\epsilon(x)}$$

$$=\frac{\epsilon(x)}{2+\epsilon(x)}$$

the limit we look for is $0$.