Evaluate the limit of f(x)-x, if x goes to -infinite and f(x)=(x^3+3x)/(x^2+1)?

We'll calculate the expression f(x) -
x:

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

f(x) - x = [(x^3+3x) -
x*(x^2+1)]/(x^2+1)

We'll remove the
brackets:

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

We'll eliminate like terms inside brackets from
numerator;

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

Now, we'll evaluate the limit, if x approaches
to -infinite:

lim [f(x) - x] = lim
(2x)/(x^2+1)

We'll force the factor x^2, at
denominator:

lim (2x)/(x^2+1) = lim 2x/x^2*(1 +
1/x^2)

We'll simplify:

lim
2/x*(1 + 1/x^2) = lim (2/x)*lim [1/(1 + 1/x^2)]

Since the
limit of 2/x approaches to zero, if x approaches to - infinite and the limit of 1/x^2
approaches to zero, if x approaches to - infinite, we'll
get:

lim (2/x)*lim [1/(1 + 1/x^2)] = 0*1/(1+0) =
0

The requested limit of the difference f(x)
- x, if x approaches to -infinite is lim [f(x) - x] =
0.