Dealing with limits that contain absolute values

Question

I've been asked to find the directional derivative of $f(x,y) = \|(x,y)\|$ at the point (0,0) in the direction of $v=(a,b)$:

$$\lim_{t\to0} \frac{\|(0,0) + t(a,b)\| - 0}{t} = \lim_{t\to0} \frac{|t|\sqrt{a^2 + b^2}}{t}$$

But why would this limit exist if the lateral limits aren't equal?

Answer 1

It's simple: it does not exist (unless $(a,b)=(0,0)$).

Answer 2

You are right -- the one-sided limits don't agree, hence the directional derivative in question does not exist.