0
$\begingroup$

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?

$\endgroup$
0
$\begingroup$

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

$\endgroup$
0
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.