Solving limit in three variables

Question

How can i solve

$$\lim_{(h,k,t)\to(0,0,0)}\frac{\sqrt{hk(z+t)}}{\sqrt{h^2+k^2+t^2}}$$

If I wanted to use polar coordinates, how can I convert the variables?

Answer 1

The limit does not exist. If you approach along a path with $h=0$ the quantity is always zero. If you approach along $h=k$ with $t=0$ the quantity is $\sqrt {\frac z2}$

Answer 2

HINT

Let use spherical coordinates with

• $h=r\sin \phi \cos \theta$

• $k=r\sin \phi \sin \theta$

• $t=r\cos \phi$

to obtain

$$\frac{\sqrt{hk(z+t)}}{\sqrt{h^2+k^2+t^2}}=\sqrt {\sin^2 \phi \sin \theta\cos \theta(z+r\cos \phi)}$$