Question

Use polar coordinates to find the limit. [if (r, ?) are polar coordinates of the point (x, y) with r ? 0, note that r ? 0+ as (x, y) ? (0, 0).] (if an answer does not exist, enter dne.) lim (x, y)?(0, 0) (x2 + y2) ln(x2 + y2)

Answer 1

`\displaystyle\lim_((x,y)\to(0,0))(x^2+y^2)\ln(x^2+y^2)=\lim_((r,\theta)\to(0^+,0))r^2\ln(r^2)=\lim_(r\to0^+)r^2\ln(r^2)`

since the limand is independent of
`\theta`.

Taking
`\rho=\frac1r`, we have (by L'Hopital's rule in the first step below)


`\displaystyle-\lim_(\rho\to\infty)(\ln\rho^2)/(\rho^2)=-\lim_(\rho\to\infty)((2\rho)/(\rho^2))/(2\rho)=-\lim_(\rho\to\infty)\frac1{\rho^2}=0`