Question

Please help me evaluate the following limit by converting to spherical coordinates.

Answer 1

Please help me evaluate the following limit by converting to spherical coordinates:
\[(\rho,\theta,\phi)\]

and by observing that:
\[\rho \rightarrow 0^{+ } \]

if and only if:
\[(x,y,z)\rightarrow(0,0,0)\]

Answer 2

\[\lim_{(x,y,z) \rightarrow (0,0,0)}\frac{ xyz }{ x^{2}+y ^{2}+z ^{2} }\]

Answer 3

In spherical coordinates you have
\[\begin{cases}
x=\rho\sin\theta\cos\phi\\
y=\rho\sin\theta\sin\phi\\
z=\rho\cos\theta\\
\rho^2=x^2+y^2+z^2
\end{cases}\]
So you have
\[\begin{align*}\lim_{(x,y,z)\to(0,0,0)}\frac{xyz}{x^2+y^2+z^2}&=\lim_{\rho\to0^+}\frac{(\rho\sin\theta\cos\phi)(\rho\sin\theta\sin\phi)(\rho\cos\theta)}{\rho^2}\\
&=\lim_{\rho\to0^+}\rho\sin^2\theta\cos\theta \sin\phi\cos\phi
\end{align*}\]