Question

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

Answer 1

Answer:

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