Question

Find the given limit lim(x,y)→(a,a)x4−y4x2−y2=lim(x,y)→(a,a) (x^4−y^4)/(x2−y2)= here aa is a constant. (find the limit assuming x≠y)

Answer 1

If $x\neq y$, then we can write

$\dfrac{x^4-y^4}{x^2-y^2}=\dfrac{(x^2-y^2)(x^2+y^2)}{x^2-y^2}=x^2+y^2$

and since this is continuous for all $x\neq y$, the limit as $(x,y)\to(a,a)$ will be

$\displaystyle\lim_{(x,y)\to(a,a)}\frac{x^4-y^4}{x^2-y^2}=\lim_{(x,y)\to(a,a)}(x^2+y^2)=2a^2$