Let $f,g : \mathbb{R^n} \to \mathbb{R}$ be differentiable at $a\in \mathbb{R^n}$.
What is the value of limit $$\lim_{x\to a} \frac{|(f(x)-f(a))(g(x)-g(a))|}{||x-a||}$$
$||n||$ just means the modulus of vector.
I tried using the definition, but I cannot go about it. I have a hint that its value has to be zero.
$\frac {|f(x)-f(a)} {\|x-a\|}$ is bounded and so is $\frac {|g(x)-g(a)} {\|x-a\|}$. Multiply these two (and multiply numerator and denominator by $\|x-a\|$) to see that the limit is necessarily $0$.