Question

If f(x,y) = {1 if (x,y) = (0,0)
sin(x,y) if (x,y)#(0,0).find lim(x,y) converges to (0,0) f(x,y)

Answer 1

f(0,0) = 1

$\lim_{(x,y) \to (0,0)} f(x,y) =\lim_{(x,y) \to (0,0)} sin(xy) = sin(0)=0.$

So limit of f(x,y) converges to 0 as (x,y) tends to (0,0).

But function is not continues at (0,0) since $\lim_{(x,y) \to (0,0)} f(x,y)\neq f(0,0)$.