- #1

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## 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.

- Thread starter KevinL
- Start date

- #1

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

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

- #2

rock.freak667

Homework Helper

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

- #3

jgens

Gold Member

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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.

- #4

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

- #5

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I dont believe you can just do that. Only if two things are multiplied beneath a radical can something be taken out.Factor out a sqrt(x^2) from the numerator.

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)?

- #6

jgens

Gold Member

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

- #7

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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]

- #8

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

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