# Dealing with limits that contain absolute values

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?

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