What is the limit of the fraction f(x)/f(x+1), if x tends to infinite and f(x)=x^3+x?

We'll calculate the denominator of the
fraction:

f(x+1) = (x+1)^3 +
x+1

f(x+1) = x^3 + 3x^2 + 3x + 1 + x +
1

We'll combine like
terms:

f(x+1) = x^3 + 3x^2 + 4x +
2

We'll evaluate the
limit:

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

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

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

Since the following limits
approaches to zero if x approaches to infinite, we'll
get:

lim f(x)/f(x+1) =
(1+0)/(1+0+0+0)

lim f(x)/f(x+1) =
1

The requested limit of the fraction
f(x)/f(x+1), if x approaches to infinite, is lim f(x)/f(x+1) =
1.