Proving definition of directional derivative $\lim_{s \to 0} \frac{f(a + sx) - f(a)}{s} = f'(a)x$

Question

Inspired by: Directional derivative question

I had long sought for a proof $$\lim_{s \to 0} \frac{f(a + sx) - f(a)}{s} = f'(a)x$$

But whenever I look it up, I always get some multivariable calculus stuff i.e. Paul's online notes http://tutorial.math.lamar.edu/Classes/CalcIII/DirectionalDeriv.aspx which is some computational multivariable calculus stuff assuming $f(x,y)$

Can someone please show that the above relation is true or provide me a reference for this formulation of the directional derivative? Thanks!

Answer 1

You can just apply L'Hopital's rule if you know $f$ is continuously differentiable.

$$\lim\limits_{s \to 0} \frac{f(a+sx)-f(a)}{s}=\lim\limits_{s \to 0} f'(a+sx)x=f'(a)x$$