Question

Find the limit, if it exists. (If an answer does not exist, enter DNE.)
lim (x, y)→(0, 0)
of (x^2 + y^2)/[((x^2 + y^2 +64)^.5)-8]

Answer 1

one good way to tackle xy limits is to convert to polar coordinates, then let r->0

f(r,theta) = r^2/(sqrt(r^2+64)-8)

now, using lHospital's Rule,

lim = 2r/[2r/(2sqrt(r^2+64))]
= 2sqrt(r^2+64))
= 2*8 = 16

Since the function does not involve theta, the limit is the same from all directions.

Answer 2

but there is no lHospitals rule in multivariable