Why is this limit 9?

[warwick]

Limit of (x^2 +18x)^(1/2) - x as x tends to infinity.

 

[skeeter]

\(\displaystyle \L [\sqrt{x^2+18x} - x] \cdot \frac{\sqrt{x^2+18x} + x}{\sqrt{x^2+18x} + x} =\)

\(\displaystyle \L \frac{(x^2 + 18x) - x^2}{\sqrt{x^2+18x} + x} =\)

\(\displaystyle \L \frac{18x}{\sqrt{x^2+18x} + x}\)

divide every term by x ...

\(\displaystyle \L \frac{18}{\sqrt{1 + \frac{18}{x}} + 1}\)

now take the limit as x -> infinity

 

[warwick]

\(\displaystyle \L [\sqrt{x^2+18x} - x] \cdot \frac{\sqrt{x^2+18x} + x}{\sqrt{x^2+18x} + x} =\)

\(\displaystyle \L \frac{(x^2 + 18x) - x^2}{\sqrt{x^2+18x} + x} =\)

\(\displaystyle \L \frac{18x}{\sqrt{x^2+18x} + x}\)

divide every term by x ...

\(\displaystyle \L \frac{18}{\sqrt{1 + \frac{18}{x}} + 1}\)

now take the limit as x -> infinity

Thanks. I didn't even think to multiply by the conjugate. I'm trying to cram for the Calculus final on Tuesday. lol.