I know that L'Hospital rule cannot be used for $2$ variable functions. But am I allowed to use it once I have converted it into polar coordinate or substitute it?
Here is an example: $f(x,y) = (x^2 + y^2)\ln(x^2+y^2)$ as $(x,y)$ approaches $(0,0)$
Here I can convert $x = r\cos(\phi)$ and $y = r\sin(\phi)$ then rewrite the function as $f(r) = \frac{\ln(r^2)}{\frac{1}{r^2}}$. Am I allowed to apply the L'Hospital rule here?
In another case, what if I substitute $x^2+y^2 = t$ then rewrite the function as $f(t) = \frac{\ln(t)}{\frac{1}{t}}$ then find the limit of $f(t)$ as $t$ approaches $0$ by applysing L'Hospital rule?