Math Help - Find the limit

1. Find the limit

Show that lim x--> infinity square root (x^2+x) -x = 1/2

note the x^2 + x is under the root only

Here's what I did so far:

[square root (x^2+x) -x] [square root (x^2+x) +x] / [square root (x^2+x) +x]

If we expand it out it becomes:

x^2+x + x[square root (x^2+x)] - x[square root (x^2+x)] - x^2 / [square root (x^2+x) +x]

1/ [square root (x^2+x) +x]

annnd that's where i lose it.. (if i ever had it from the start)

2. Originally Posted by alisheraz19
Show that lim x--> infinity square root (x^2+x) -x = 1/2

note the x^2 + x is under the root only

Here's what I did so far:

[square root (x^2+x) -x] [square root (x^2+x) +x] / [square root (x^2+x) +x]

If we expand it out it becomes:

x^2+x + x[square root (x^2+x)] - x[square root (x^2+x)] - x^2 / [square root (x^2+x) +x]

1/ [square root (x^2+x) +x]

annnd that's where i lose it.. (if i ever had it from the start)

The last expression is wrong: it must have x in the numerator. No wonder, since you expanded the product in the numerator in a terrible way: didn't you pay attention to the fact that you had an expression of the very easy form $(a-b)(a+b)=a^2-b^2=$ difference of squares?
Fix this, and then multiply the whole expression by $1=\frac{\frac{1}{x}}{\frac{1}{x}}$ so that you'll be able to divide each term in numerator and denominator by x (and get the x into the square root as $x^2$...!)

Tonio

3. I'm not quite following, the expression I have so far is :

x / square root (x^2+x) + x note* x^2+x is under the root only

i don't understand how to get 1/2 from here.. i expanded it out that way because i'm not very good with remembering rules ..

where do i go from here? how does multiplying by 1 = 1/x / 1/x ?

4. Originally Posted by alisheraz19
I'm not quite following, the expression I have so far is :

x / square root (x^2+x) + x note* x^2+x is under the root only

i don't understand how to get 1/2 from here.. i expanded it out that way because i'm not very good with remembering rules ..

where do i go from here? how does multiplying by 1 = 1/x / 1/x ?
$\frac{1/x}{1/x}\cdot\frac{x}{\sqrt{x^2+x}+x}=\frac{1}{\frac{1}{ x}(\sqrt{x^2+x}+x)}=\frac{1}{\sqrt{\frac{x^2+x}{x^ 2}}+1}$

Do you understand what to do from here?