- #1

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**A limit problem!!**

I was trying to calculate the following limit:

lim {x-> infinity} (x² + 2x)^(1/2) - x

I manipulate f(x) in such a way:

f(x) = (x² + 2x)^(1/2) - x

f(x) = [x²(1 + 2/x)]^(1/2) - x

f(x) = |x|(1 + 2/x)^(1/2) - x

As x goes to infinity |x| = x. Therefore,

f(x) = x( [1 + 2/x]^(1/2) - 1 ]

After taking the limit I'm getting infinity times 0 which would mean the limit is equal to zero (I think). However, this isn't the right answer and the way I was supposed to do it was multiply f(x) by its "conjugate" to get the limit as equal to 1. But, I've tried figuring out why I can't do the above, but I just don't see it. So I'd appreciate the help.