Calculate the limit of the function y=x^2/(x^2+1), if x --> +infinte?

Since = and
lim_(x->oo) x^2

We can use
L'Hopital's rule that =

if f(x) = and
g(x) =

So
(x^2)/(x^2+1) = 2x/2x = 1 =
1

So (x^2)/(x^2+1) =
1

We could also use the following analysis.  Using the fact
that

1/x^n = 0, and dividing both the
denominator and numerator by
x^2 (x^2/x^2 = 1 and (x^2+1)/x^2= 1 + 1/x^2) we
get

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

and again since
1/x^2 = 0 we get

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

This second method also shows
that

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