Identify all horizontal and vertical asymptotes of the graph of the function. f(x)=x^2/x^2-4

Question

I know the vertical asymptotes are x=2 and x=-2, but how do I find the horizontal asymptotes?

Answer 1

Just consider what happens when x gets large. At x = 10^100, say, all but the highest powers of x get swamped, so f(x) just looks like

x^2/x^2 = 1

So, y=1 is the horizontal asymptote

Or, you can do a polynomial division, to see that

x^2/(x^2-4) = 1 + 4/(x^2-4)

as x gets huge, 4/(x^2-4) goes to zero, leaving f(x) = 1.