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

[KevinL]

Homework Statement

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

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]

use the fact that $x= \sqrt{x^2}$

 

[jgens]

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]

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

 

[KevinL]

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]

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

 

[JG89]

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)?

$$\sqrt(1+3x^2) = \sqrt(x^2(\frac{1}{x^2} + 3)) = \sqrt(x^2) \sqrt(\frac{1}{x^2} + 3)$$

 

[KevinL]

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