Limit of Function with Log: What is the Limit of f(x) as x Approaches Infinity?

[MaxManus]

Homework Statement

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

x-> ∞

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?

 

[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