Could anyone please help me proving by definition that:
lim_(x,y)-->(2,1) f(x,y)=3
where f(x,y)= (x+y) / (x-y)
Thank you in advance.
The numerator and denominator are both continuous functions, so the only place where $\displaystyle f(x, y)$ is not continuous is where the denominator $\displaystyle = 0$. This does not occur at $\displaystyle (x, y) = (2, 1)$, so you can just use direct substitution.
$\displaystyle \lim_{(x, y) \to (2, 1)}\frac{x + y}{x - y} = \frac{2 + 1}{2 - 1}$
$\displaystyle = \frac{3}{1}$
$\displaystyle = 3$.