Limit and Continuity of Functions of Two Variables

Evaluate `lim_((x,  y) -> (1,  2))  "g"(x, y)`, if the limit exists, where `"g"(x, y) = (3x2 - xy)/(x^2 + y^2 + 3)`

Evaluate `lim_((x,  y) -> (0,  0)) cos((x^3 + y^2)/(x + y + 2))` If the limits exists

Let f(x, y) = `(y^2 - xy)/(sqrt(x) - sqrt(y))` for (x, y) ≠ (0, 0). Show that `lim_((x,  y) -> (0,  0)) "f"(x,  y)` = 0

Evaluate `lim_((x,  y) -> (0,  0)) cos(("e"^x sin y)/y)`, if the limit exists

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)` = 0 along every line y = mx, m ∈ R

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}

Show that f(x, y) = `(x^2 - y^2)/(y - 1)` s continuous at every (x, y) ∈ R² 

Let g(x, y) =  `("e"^y  sin x)/x` for x ≠ 0 and g(0, 0) = 1 shoe that g is continuous at (0, 0)