Homework Help: Find the limit as x approaches infinity of (sqrt(1+3x^2))/x

KevinL · Feb 24, 2009

1. The problem statement, all variables and given/known data
Find the limit as x approaches infinity of (sqrt(1+3x^2))/x

3. The attempt at a solution
I tried using l'hopital's rule, but it gave me 3x/(sqrt(1+3x^2)) which doesnt help me at all.

rock.freak667 · Feb 24, 2009

Re: limit

use the fact that [itex]x= \sqrt{x^2}[/itex]

jgens · Feb 24, 2009

Re: limit

Well, I can think of a couple of ways to do this one. Probably the simplest is to note that for arbitrarily large x, 3x^2 + 1 ~ 3x^2. Another way would be to multiply and divide the equation by x and then try to find the limit.

JG89 · Feb 24, 2009

Re: limit

Factor out a sqrt(x^2) from the numerator.

KevinL · Feb 24, 2009

Re: limit

JG89 said: ↑

Factor out a sqrt(x^2) from the numerator.

I dont believe you can just do that. Only if two things are multiplied beneath a radical can something be taken out.

Multiplying by x/x does not help either.

Hmm, perhaps you can say that as x gets arbitrarily large, its sqrt(3x^2)/x because the 1 becomes meaningless and thus the limit is sqrt(3)?

jgens · Feb 24, 2009

Re: limit

Yes you can do that! However, multiplying and dividing by x certainly does help: Lim (x -> infinity) sqrt(1/x^2 + 3)/1.

JG89 · Feb 24, 2009

Re: limit

KevinL said: ↑

I dont believe you can just do that. Only if two things are multiplied beneath a radical can something be taken out.

Multiplying by x/x does not help either.

Hmm, perhaps you can say that as x gets arbitrarily large, its sqrt(3x^2)/x because the 1 becomes meaningless and thus the limit is sqrt(3)?

[tex]\sqrt(1+3x^2) = \sqrt(x^2(\frac{1}{x^2} + 3)) = \sqrt(x^2) \sqrt(\frac{1}{x^2} + 3)[/tex]

KevinL · Feb 24, 2009

Re: limit

Ah, clever. Thank you. Thats definitely how my prof would want me to do it.