Limit of $f(x,y)=(x^3-y^3)/(x-y)$ as (x,y) goes to (1,1)

Question

Find the limit of $f(x,y)=(x^3-y^3)/(x-y)$ as (x,y) goes to (1,1)

I approached (1,1) along the x-axis, y-axis and the line x=y, I got 3 as a limit each time, so I assume the limit is 3. Next, I have to prove the limit using the $\epsilon$ $\delta$ definition.

What I got: Let $\epsilon,\delta>0$ such that $0<\sqrt{(x-1)^2+(y-1)^2}<\delta$

Then, $|(x^3-y^3)/(x-y)-3)|=|(x^3-y^3-3x+3y)/(x-y)|$

Then I am blocked here, don't know what to compare the above equation to.

Answer 1

Note that $$ \lim_{(x,y)\to(1,1)}\frac{x^3-y^3}{x-y}=\lim_{(x,y)\to(1,1)}\frac{(x-y)(x^2+xy+y^2)}{x-y}=\lim_{(x,y)\to(1,1)}(x^2+xy+y^2)=3 $$