Limit's solving method

Question

If $\lim_{x\to 0}{(\frac{f(x)}{x})}=a\neq0$, then find $\lim_{x\to 0}{[\frac{f(ax)}{x^2+x}]}$. Any ideas for the solving method?

Answer 1

You can do the substitution $t=ax$, so the limit becomes $$ \lim_{t\to0}\frac{f(t)}{\dfrac{t^2}{a^2}+\dfrac{t}{a}} = a^2\lim_{t\to0}\frac{f(t)}{t^2+at} $$ Now it should be easy.

Answer 2

Replacing $x$ by $ax$ everywhere on the LHS, you can write the first limit also (since $a \neq 0 $) as

$$ \lim_{x\to 0}{(\frac{f(ax)}{ax})}=a\neq0 $$

So you know that near $x=0$, $f(ax) \simeq a^2 x$.

Inserting this into the second limit gives

$$ \lim_{x\to 0}{[\frac{f(ax)}{x^2+x}]} = \lim_{x\to 0}\frac{a^2 x}{x^2+x} = \lim_{x\to 0}\frac{a^2}{x+1} = a^2 $$