Answer to Question #135763 in Calculus for Promise Omiponle

Question

3. Evaluate the limit

lim(x,y)->(2,2) (x-y)/(x^4-y^4)

Answer 1

$x^4-y^4=(x^2)^2-(y^2)^2$

Using the formula $(a^2-b^2)=(a+b)(a-b),$ we get

$x^4-y^4=(x^2+y^2)(x^2-y^2)$

Again, $(x^2-y^2)$ can be written as $(x+y)(x-y)$

$\therefore x^4-y^4=(x^2+y^2)(x+y)(x-y)$

Now $\lim_{(x,y)\rightarrow(2,2)}\frac{x-y}{x^4-y^4}$

$=\lim_{(x,y)\rightarrow(2,2)}\frac{x-y}{(x^2+y^2)(x+y)(x-y)}$

$=\lim_{(x,y)\rightarrow(2,2)}\frac{1}{(x^2+y^2)(x+y)}$

$=\frac{1}{(2^2+2^2)(2+2)}$

$=\frac{1}{32}$

$\therefore \lim_{(x,y)\rightarrow(2,2)}\frac{1}{(x^2+y^2)(x+y)}=\frac{1}{32}$