If $\lim_{x \to 0} f(x)/x = 1$, then $\lim_{x \to 0} f(x)=0$ This statement is said to be true, but I am not sure how to illustrate it.
I found an old thread that discusses using the product limit law.
Just intuitively I am struggling as well.
The only function for $f(x)$ I can think of that would make $\lim_{ x\to 0} f(x)/x = 1$ is $x$. Therefore $x/x=1$
So if we know the limit goes to $1$ then that means we know that the function $f(x)$ must be cancelled to make a $1$.
Furthermore I thought we couldn't have $0/0$?
How do I look at this problem?

I am wondering whether the title should say "lim x -> 0 f(x) = 0"? I will assume that this is what you meant. As you mention, it does follow from the fact that a product of limits is equal to the limit of the product whenever the first two limits exist. In your case,
\begin{align*}
\lim_{x \to 0} f(x) = 
\lim_{x \to 0} \frac{f(x)}{x} \cdot x =
\left(\lim_{x \to 0} \frac{f(x)}{x}\right)\left(\lim_{x \to 0} x\right) = 1 \cdot 0 = 0
\end{align*}
as required.
