5
$\begingroup$

Given $\frac{a}{c} + \frac{b}{d} = 1$ Prove that $$\lim_{(x,y)\rightarrow(0,0)} \frac{|x|^{a}|y|^{b}}{|x|^{c} + |y|^{d}}$$ does not exist.

So I have done the proof for strict inequalities. And the limit only exists when the fractions add up to an integer greater than $1$. I'm not sure how to approach this other than counter examples.

$\endgroup$
4
$\begingroup$
  1. Taking the limit along the axes gives zero.
  2. Taking $|x|^c=|y|^d=t\to 0$ gives $$ \frac{t^\frac{a}{c}\cdot t^\frac{b}{d}}{t+t}=\frac{t}{2t}=\frac12. $$
$\endgroup$

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.