How to compute this limit
$$\mathop {\lim }\limits_{n \to \infty } \left( {n - {n^2}\ln \left( {1 + {1 \over n}} \right)} \right)$$
I took it from an exem. According to the solution, it's not $0$. I have no idea how to proceed. Thanks for the help.
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2What have you done so far ? – Nov 20 '15 at 15:37
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I don't know how to add an image to here, can u help me? – Yarden Sharabi Nov 20 '15 at 15:52
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You can write it as $\frac {n^3-n^4 \ln (1+1/n)}{n^2}$ and useL'Hospital. The reason for the $n^2$ is hard to motivate without Taylor's theorem, though. – Ian Nov 20 '15 at 15:57
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Hint:
Use Taylor's formula at order $2$ for $\;\ln\Bigl(1+\frac1n\Bigr)$. You should find $\dfrac12$.
Another hint, without Taylor's formula:
Use L'Hospital's rule to find the limit of $\;\dfrac{x-\ln(1+x)}{x^2}$ when $x$ tends to $0$, then set $\;x=\dfrac1n$.
Bernard
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we didn't learn about taylor yet, can't find something more simple? – Yarden Sharabi Nov 20 '15 at 15:50
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1Then, state in your question what you have done, what you have learned, what you can and cannot use. That is relevant. – Clement C. Nov 20 '15 at 15:54
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