Use spherical coordinates to find the limit. [Hint: Let x = rho sin (ϕ) cos (θ), y = rho sin (ϕ) sin (θ), and z = rho cos (ϕ), and note that (x, y, z) → (0, 0, 0) implies rho → 0+.] lim (x, y, z) → (0, 0, 0) arctan 19 x2 + y2 + z2

lim (x, y, z) → (0, 0, 0) [x*y*z] / (x^2+y^2+z^2) = 0

Step-by-step explanation:

First, we need to put the spherical coordinates equations that we will use in our problem

x = p*sin∅cos Ф

y = p*sin∅sinФ

z=p*cos∅

Then we will state the problem

lim (x, y, z) → (0, 0, 0) [x*y*z] / (x^2+y^2+z^2)

Using the spherical coordinates we get

(x^2+y^2+z^2) = p^2

Which will make our limit be

lim p→0 + [p*sin∅cos Ф*p*sin∅sinФ * p*cos∅] / (x^2+y^2+z^2)

after solving limit:

lim (x, y, z) → (0, 0, 0) [x*y*z] / (x^2+y^2+z^2) = 0