Question

hi, I'm trying to find out whether it is possible to make the function x*ln(x^2+3y^2) continuous at (0,0). I've tried the squeeze theorem to prove that the limit is 0 (according to the textbook) didn't succeed. I'll appreciate your help. thanks

Answer 1

To make a function continuous for some \((x,y)=(a,b)\), you have to establish that
\[(1)~~\lim_{(x,y)\to(a,b)}f(x,y)=f(a,b)\\
(2)~~f(a,b)\text{ exists}\]
The second condition isn't met, but you since you have the limit as 0, you can change \(f(x,y)\) to say
\[f(x,y)=\begin{cases}x\ln(x^2+3y^2)&\text{for }(x,y)\not=(0,0)\\
0&\text{for }(x,y)=(0,0)
\end{cases}\]
Not sure if this is the answer you're looking for...