Indeterminant forms homework help

[EngnrMatt]

Homework Statement

lim{x\rightarrow∞} sqrt(x^(2)+5x+11)-x

Homework Equations

I know it is of type ∞-∞

The Attempt at a Solution

I have worked this problem around to death, and I know I'm supposed to give them a common denominator to get ∞/∞ and use L'Hospital's Rule, but I end up with ∞-∞ in the numerator or the denominator every time. Professor never went over problems this complex in class, and now we are too far ahead of this section to ask her to waste time going back to it.

I know the answer is 2.5 (graphing calculator), but my problem is figuring out how to work it.

 

[Dick]

Try multiplying and dividing by sqrt(x^(2)+5x+11)+x. It's the 'conjugate' trick. It will turn out you don't really even need l'Hopital. You can do it all with algebra.

 

[CAF123]

You are right you can use i'Hopital but after one differentiation, use: $$\sqrt{x^2 + 5x + 11} = \sqrt{x^2} \sqrt{1 + 5/x + 11/x^2}, $$ simplify and cancel terms and then take x → $\infty$.

 

[EngnrMatt]

Thanks guys! I knew it had something to do with the conjugate, but for some reason I had trouble remembering just how it worked. I've worked it out now though, thanks for the help!