Evaluate the limit: $\lim_{(x,y,z)\to(0,0,0)}\frac{xy+yz^2+xz^2}{x^2+y^2+z^4}$

Question

Could someone give me a hint? I would like to continue to attempt it. The limit to evaluate that I would like a hint on is:

$$\lim_{(x,y,z)\to(0,0,0)}\frac{xy+yz^2+xz^2}{x^2+y^2+z^4}$$

Answer 1

1. Along the curve $x=y=z^2$, the function is identically $1$.

2. Along the line $y=z=0$, the function is identically $0$.

3. Along the line $x=y, z=0$, the function is identically $\frac 12$.

4. Along the line $x=2y, z=0$, the function is identically $\frac 25$.