# Homework Help: Limit of function with log

1. Nov 16, 2011

### MaxManus

1. The problem statement, all variables and given/known data

Find the limit of
f(x) = x^2 ln(1+1/x) - x

x-> ∞

3. The attempt at a solution

Not sure where to start, but I take the derivative

f'(x) = $2x*ln(1+1/x) +x^2 \frac{ -x^{-2}}{1+1/x} -1$
$2x*ln(1+1/x) - \frac{1}{1+1/x} -1$

the second term goes to -1 as x->∞ ant the last term is always -1. Can I say somethong about the last term and will it help me?

2. Nov 16, 2011

### MaxManus

Got it.

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

= $\frac{ln(1+1/x) - (1/x)}{1/x^2}$

u = 1/x

f(u) = (ln(1+u) -u)/u^2

which limit is -1/2 when u->0

The limit is found by using l hopital twice

Last edited: Nov 16, 2011
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