limit x/(x+1)^2 x--> inf.

First let us open the brackets:

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

Now let us divide by the highest power of the function which is x^2

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

Now substitute when x--> inf.

==> lim 0/1+0+0 = 0/1 =0

Then the limit when x--> infinity is 0.

To evaluate Lt x/(x+1)^2 as x-->infinity.

Solution:

x/(x+1)^2 = {(x+1)-1}/(x+1)^2

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

Therefore,

Lt x-->inf {x/(x+1)^2} = Lt x-->inf [1/(x+1)]- Lt x--> inf [(1/(x+1)^2]

= 0-0

= 0.

First, we'll expand the square from the denominator, using the formula (a+b)^2=a^2+2ab+b^2.

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

In order to calculate the limit of a rational function, when x tends to +inf., we'll divide both, numerator and denominator, by the highest power of x, which in this case is x^2.

We'll have:

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

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

After reducing similar terms, we'll get:

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