Answer to Question #84929 in Calculus for Dillip

Question

The asymptotes of the curve
x^2y^2 = x^2-y^2 are y = ±1.

Is the statement true or false?
Give a short proof or a counter example in
support of your answer.

Answer 1

The statement is true.

x^2*y^2=x^2 - y^2 <=> y^2 = x^2/(x^2 + 1),

y=x/sqrt(x^2+1) or y=-x/sqrt(x^2+1),

x^2/(x^2 + 1)->1 as x -> +infinity, x^2/(x^2 + 1)->1 as x -> -infinity, 

y ->1 as x -> infinity or y ->-1 as x -> infinity.