# Inhomogeneous fractional limit with three variables

To evaluate the following limit:

$$\lim _ { ( x , y , z ) \rightarrow ( 0 , 0 , 0 ) } \frac { x ^ { 2 } + y ^ { 4 } + z ^ { 6 } } { x + y ^ { 2 } + z ^ { 3 } }$$

Is it reasonable to set $$x=y^5$$ and $$z=-y^{\frac{2}{3}}$$ and draw the conclusion that the limit does not exist?

I also tried to use the spherical coordinate and get $$\frac{0}{\sin(\phi)\cos(\theta)}$$ which I'm not sure how to deal with?

• Your substitution appears to support the conclusion that the limit is $0$. Did you try a second approach curve and get a different result? What other evidence leads you to believe that the limit does not exist? – abiessu Feb 23 at 14:57