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Calculus/limit at infinity with radical

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Expert: - 4/30/2010


Question
find the limit as x approaches infinity of:
sqrt(x^2+9))/sqrt(9x^2+1)
I know the answer is 1/3, but I just can't get there. I've tried l'Hopital's rule, but the limit to infinity is confusing me.
Thanks for your help.

Answer
Since we ave √(x²+9)/√(9x²+1), that is the same as √[(x²+9)/(9x²+1)].

If we divide the numerator and the denominator by x², we get √[(1 + 9/x²)/(9 + 1/x²).

If we let x->∞, then for both 9/x² and 1/x² we get 0.
So the lim x->∞ √[(1 + 9/x²)/(9 + 1/x²) =  √[(1 + lim(x->∞)[9/x²])/(9 + lim(x->∞)[1/x²]).

Combining both of the previous statements leaves √(1+0)/(9+0) = √(1/9) = 1/3.


When trying to compute the limits of one polynomial in x / (another ploynomial in x)
as x -> ±∞, divide the top and bottom with x^m where m is the greatest power that is seen.

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