Finding Limits using a Taylor Series

**adam63** · March 20th 2010, 06:14 AM

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

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

Thank you very much

**CaptainBlack** · March 20th 2010, 06:40 AM

Originally Posted by adam63

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

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

Thank you very much

Put $u=1/x$ , then you want:

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

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

CB

**adam63** · March 20th 2010, 07:11 AM

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 $x^2$ 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 ...

**adam63** · March 21st 2010, 01:33 AM

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?

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