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))(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))(7e^(-r^2)-7)/(r^2)`

Most important is that the limand is independent of
`t`. Evaluating directly yields
`\frac00`, so we apply L'Hopital's rule:


`\displaystyle\lim_(r\to0)(7e^(-r^2)-7)/(r^2)=\lim_(r\to0)(-14re^(-r^2))/(2r)=-7\lim_(r\to0)e^(-r^2)=-7`