Find the limit, if it exists, or show that the limit does not exist. lim of x+y/y^2-x^2 as (x,y) approaches the point (-1,1).

Answer 1

Evaluate the limit along the line y=mx where m∈R.
f(x,y)
=(x+y)/(y^2-x^2)
=(x+mx)/(m^2x^2-x^2)
=x(1+m)/[x^2(m+1)(m-1)]
=1/[(m-1)x]

Since the limit depends on m, the limit does not exist.