Question

lim (x,y,z)->(0,1,0) x^3y^2+y^2z^5/x^3z^5

Answer 1

$L=\lim\limits_{(x,y,z)\to(0,1,0)}\dfrac{x^3y^2+y^2z^5}{x^3z^5}\\ \\ \\ \text{Dominio de }\dfrac{x^3y^2+y^2z^5}{x^3z^5} \text{ e } \left\{(x,y,z)\in \mathbb R^3:x\neq 0, z\neq 0 \right\}\\ \\ \\ L=\lim\limits_{(x,y,z)\to(0,1,0)}\dfrac{(x^3+z^5)y^2}{x^3z^5}$

$A=\{(mt,t+1,nt) ,m\neq 0, n\neq 0,t\in \mathbb R\}\subset \left\{(x,y,z):x\neq 0, z\neq 0 \right\}\\ \\ \text{Ent\~ao quando : }(x,y,z)\in A\\ \\ L=\lim\limits_{t\to0}\dfrac{(m^3t^3+n^5t^5)(t+1)^2}{m^3n^5t^8}\\ \\ \\ L=\lim\limits_{t\to0}\dfrac{(m^3+n^5t^2)(t+1)^2}{m^3n^5t^5}\\ \\ \\ L=\lim\limits_{t\to0}\dfrac{1}{n^5t^5}$

Entonces el límite no existe