# Thread: Finding Limits using a Taylor Series

1. ## Finding Limits using a Taylor Series

I need to find :
$\displaystyle \lim_{x \rightarrow \infty} x-x^{2}ln(1+\frac{1}{x})$

Only I have to use a Taylor Series here, and I mustn't use L'Hopital's rule.

Thank you very much

I need to find :
$\displaystyle \lim_{x \rightarrow \infty} x-x^{2}ln(1+\frac{1}{x})$

Only I have to use a Taylor Series here, and I mustn't use L'Hopital's rule.

Thank you very much
Put $\displaystyle u=1/x$, then you want:

$\displaystyle \lim_{u \to 0} \left[ \frac{1}{u}-\frac{1}{u^2}\ln(1+u) \right]$

Now expand $\displaystyle \ln(1+u)$ as a Taylor series about $\displaystyle u=0$ as far as the term in $\displaystyle u^3$ simplify and evaluate the limit.

CB

3. Thank you very much

BTW - Somebody posted a very practical reply, that said I need to call f(x)=ln(1+x), and use taylor there, then put (1/x) instead of (x), and muliplying it by $\displaystyle x^2$ and deducting it from x will give me the exact answer. This way seems very nice and 'to the point', but I wonder why the one who wrote it decided to delete it :O ...

4. That way, of expanding ln(1+x), then putting (1/x) instead of (x), and multiply it by x^2 and all that is a proper proof, isn't it?