<s>[INST] Using $\epsilon$-$\delta$ approach to prove that $\lim_{(x,y,z)\rightarrow (0,0,0)}\frac {y^3-1000xy^2+z^5}{x^2+y^2+z^4}=0$? As the title indicates, Using $\epsilon$-$\delta$ approach to prove that $$\lim_{(x,y,z)\rightarrow (0,0,0)}\frac {y^3-1000xy^2+z^5}{x^2+y^2+z^4}=0$$?
 [/INST] Here is a method without spherical coordinates, only crude inequalities. First, we have:
$$\left| \frac{y^3-1000xy^2+z^5}{x^2+y^2+z^4} \right| \leq \frac{|y|^3}{x^2+y^2+z^4}+1000\frac{|x|y^2}{x^2+y^2+z^4}+\frac{|z|^5}{x^2+y^2+z^4}.$$
Since each of $x^2$, $y^2$ and $z^4$ is non-negative, we have $x^2+y^2+z^4 \geq y^2$ and $x^2+y^2+z^4 \geq z^4$, so that:
$$\left| \frac{y^3-1000xy^2+z^5}{x^2+y^2+z^4} \right| \leq \frac{|y|^3}{y^2}+1000\frac{|x|y^2}{y^2}+\frac{|z|^5}{z^4} = |y|+1000|x|+|z|.$$
Hence, the limit at $(0,0,0)$ of this function is $0$.
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