Question

Let g(x, y) = `(x^2y)/(x^4 + y^2)` for (x, y) ≠ (0, 0) = 0. Show that `lim_((x,  y) -> (0,  0)) "g"(x,  y) = "k"/(1 + "k"^2)` along every parabola y = kx², k ∈ R\{0}

Answer 1

Given g(x, y) = `(x^2y)/(x^4 + y^2)` for (x, y) ≠ (0, 0) and f(0, 0) = 0

 g(x, y) = `(x^2y)/(x^4 + y^2)` along every parabola y = kx², k ∈ R\{0}

`lim_((x,  y) -> (0,  0)) "g"(x, y) = lim_((x,  kx^2) -> (0,  0)) (x^2(kx^2))/(x^4 + (kx^2)^2`

= `lim_((x,  kx^2) -> (0,  0)) (kx^4)/(x^4 (1 + k^2)^2`

= `lim_((x,  kx^2) -> (0,  0)) ((k)/(1 + k^2))`

= `k/(1 + "k"^2)`

Hence proved.