Limit defined (?) I wonder, if $ \lim\limits_{x \to 0} {f(x) \over x} = 0 $ (especially, if it is well defined.) 
If the function f(x)= 0 for all x apart from finitely many. 
As my understandings goes: $ {f(x) \over x} $ doesn't behave any differently than {0/x}, when x tends to infinity.. (which gets me the above result)
Another question: How do I get rid of my self insecurness in mathematics? Every time I reach a result, be at a "minor" or a "bigger" one, I doubt myself as I am always afraid of messing up somewhere on the road. 

If $f(x)=0$ for all but finitely many values of $x$, then $f(x)=0$ for all $x$ in a small punctured neighborhood of $x=0$. Hence
$$
\lim_{x \to 0} \frac{f(x)}{x} = \lim_{x \to 0} \frac{0}{x} =0.
$$
For the second question, sorry but I am a mathematician ;-)
