How do you find the slant asymptote of y=(x^3)/((x^2)-3)?

1 Answer
Aug 14, 2018

The slant asymptote is: y=x

Explanation:

Given:

y = x^3/(x^2-3)

Note that:

x^3/(x^2-3) = (x^3-3x+3x)/(x^2-3)

color(white)(x^3/(x^2-3)) = (x(x^2-3)+3x)/(x^2-3)

color(white)(x^3/(x^2-3)) = x+(3x)/(x^2-3)

and:

lim_(x->+-oo) (3x)/(x^2-3) = lim_(x->+-oo) 3/(x-3/x) = lim_(x->+-oo)3/x = 0

So:

y = x^3/(x^2-3)

is asymptotic to y=x