Question

Use polar coordinates to find the limit. [if (r, θ) are polar coordinates of the point (x, y) with r ⥠0, note that r â 0+ as (x, y) â (0, 0).] (if an answer does not exist, enter dne.) lim (x, y)â(0, 0) 7eâx2 â y2 â 7 x2 + y2

Answer 1

I'll take a wild guess and suggest the limit is supposed to be

$\displaystyle\lim_{(x,y)\to(0,0)}\frac{7e^{-x^2-y^2}-7}{x^2+y^2}$

Converting to polar coordinates, we take $x^2+y^2=r^2$ with $x=r\cos t$ and $y=r\sin t$. This yields

$\displaystyle\lim_{(r,t)\to(0,0)}\frac{7e^{-r^2}-7}{r^2}$

Most important is that the limand is independent of $t$. Evaluating directly yields $\dfrac00$, so we apply L'Hopital's rule:

$\displaystyle\lim_{r\to0}\frac{7e^{-r^2}-7}{r^2}=\lim_{r\to0}\frac{-14re^{-r^2}}{2r}=-7\lim_{r\to0}e^{-r^2}=-7$