$\text { Use spherical coordinates to find the limit } \lim _ { ( x , y , z ) \rightarrow ( 0,0,0 ) } \frac { 9 x y z } { 9 x ^ { 2 } + 9 y ^ { 2 } + 9 z ^ { 2 } } \text {. }$ (Hint: Let $x = \rho \sin \phi \cos \phi , y = \rho \sin \phi \sin \theta$, and $z = \rho \cos \phi$, and note that $( x , y , z ) \rightarrow ( 0,0,0 )$ implies $\left. p \rightarrow 0 ^ { + } . \right)$
A) 6
B) 9
C) 1
D) 7
E) 0

Question

Quizplus · Accepted Answer

In spherical coordinates, $x = \rho \sin \phi \cos \theta$, $y = \rho \sin \phi \sin \theta$, and $z = \rho \cos \phi$. Substituting these into the given expression, we get $\frac{9\rho^3 \sin^2\phi \cos\phi \sin\theta \cos\theta}{9\rho^2}$. Simplifying, we get $\rho \sin^2\phi \cos\phi \sin\theta \cos\theta$. As $\rho \rightarrow 0$, the entire expression approaches 0, regardless of the values of $\phi$ and $\theta$.

(Hint: Let x=ρsin⁡ϕcos⁡ϕ,y=ρsin⁡ϕsin⁡θx = \rho \sin \phi \cos \phi , y = \rho \sin \phi \sin \thetax=ρsinϕcosϕ,y=ρsinϕsinθ

(Hint: Let $x = \rho \sin \phi \cos \phi , y = \rho \sin \phi \sin \theta$