Find the slant asymptote of f(x)=x^3/(x+2)^2

The equation of the oblique (slant) asymptote
is:

y = mx + n

We need to
determine m and n to find the equation of the slant
asymptote.

m = lim f(x)/x, if x approaches to
+infinite

m = lim
x^3/x*(x+2)^2

We'll expand the square from
denominator:

lim x^3/x*(x+2)^2 = lim x^3/x(x^2 + 4x +
4)

lim x^3/x(x^2 + 4x + 4) = lim x^3/(x^3 + 4x^2 +
4x)

We'll force the factor x^3 at
denominator:

lim x^3/x^3(1 + 4/x + 4/x^2) = lim 1/(1 + 4/x
+ 4/x^2)

lim 1/(1 + 4/x + 4/x^2) = lim 1/(1 + lim4/x +
lim4/x^2)

lim 1/(1 + lim4/x + lim4/x^2) = 1/(1+0+0) =
1

Since m = 1 and it is a finite value, we'll calculate
n:

n = lim [f(x) - mx] = lim [x^3/(x+2)^2 -
x]

lim [x^3/(x+2)^2 - x] = lim (x^3 - x^3 - 4x^2 - 4x)/(x^2
+ 4x + 4)

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

The equation of the slant asymptote, if x
approaches to + infinite and - infinite, is y = x -
4.