Find the limit of $f(x)$ given the limit of $\frac{f(x)}{x}$ Given that:

$$\lim_{x\to 0}\frac{f(x)}{x}=1.$$

How do I evaluate $$\lim_{x\to 0}f(x)$$ and $$\lim_{x\to 0}\frac{f(x)}{g(x)} $$ given $\lim_{x\to 0}g(x)$ exists and is non zero.
Hint: $\lim_{x\to 0}{f(x)} = \lim_{x\to 0}{x\cdot \frac{f(x)}{x}}$.  What do you know about limits of products or quotients?
It's not a property, you are just multiplying and dividing a function by a number that is not zero(even though it tends to 0), so you are not changing the function.