What is the limit of
$$f(x,y) = (x^2+y^2)\log|x+y|$$
as $(x,y) \rightarrow (0,0)$
I tried using Squeeze Theorem but couldn't think of an upper bound for it. I know the limit of $t log(t)$ is 0 as $t \rightarrow 0$.
$\endgroup$2 Answers
$\begingroup$The limit does not exist. Indeed, $f(x,x)=x^2\log (4x^2)$ tends to $0$ as $x\to 0$ (using $t=4x^2$ in what you wrote), whereas $f(x,-x)=2x^2\log 0=-\infty$ for all $x$ (and in particular as $x\to 0$).
$\endgroup$ 12 $\begingroup$Could you not Equate x to y since they approach the same value?
*log%282x%29&var=x&val=0&dir=&steps=on
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