Limit finding with auxiliary function

Question

I try to find the $\lim_{x\to 0}{g(x)}$, when $g(x)=\frac{2f(x)+x}{f(x)+2x}$ and $\lim_{x\to 0}{f(x)}=0$. I assume that I can't say $\lim_{x\to 0}{g(x)}=lim_{x\to 0}{\frac{2f(x)+x}{f(x)+2x}}$ and replace the value of $\lim_{x\to 0}{f(x)}$, because I don't know if $\lim_{x\to 0}{g(x)}$ actually exists. So how can I find the $\lim_{x\to 0}{g(x)}$?

Answer 1

$$g (x)=\frac {2\frac {f (x)}{x}+1}{\frac {f (x)}{x}+2} $$

so if $\lim_0f'(x)=L $ then $$\lim_0g (x)=\frac {2L+1}{L+2} $$

Answer 2

In general $\lim_{x\to 0}{g(x)}$ does not exist. To this end consider $f(x)=|x|$. Then

$\lim_{x\to 0+0}{g(x)}=1$ but $\lim_{x\to 0-0}{g(x)}=-1$