find the limit (if it exists). (if an answer does not exist, enter dne.)\n\\(\\lim_{(x,y)\\to(0,0)}\\ln(x^{2}…
find the limit (if it exists). (if an answer does not exist, enter dne.)\n\\(\\lim_{(x,y)\\to(0,0)}\\ln(x^{2}+y^{2})\\)
Answer
Explanation:
Step1: Convert to polar coordinates
Let $x = r\cos\theta$ and $y = r\sin\theta$. Then $x^{2}+y^{2}=r^{2}$, and as $(x,y)\to(0,0)$, $r\to0$. The limit becomes $\lim_{r\to0}\ln(r^{2})$.
Step2: Simplify the expression
$\lim_{r\to0}\ln(r^{2})=\lim_{r\to0}2\ln(r)$. As $r\to0^{+}$, $\ln(r)\to-\infty$, so $2\ln(r)\to-\infty$.
Answer:
$-\infty$