# Thread: How could I solve this limit with Taylor series?

1. ## How could I solve this limit with Taylor series?

Hello everyone.

I am not sure how to solve this with Taylor/Mc-Laurin series:

$\frac{log(e^x-x-x^2)}{logsinx-logx}$

I have no problems at the numerator, I mean, I could do Mc-Laurin series for e^x so then I'd have a "1" which could help out in the following Mc-Laurin development:

log(1+x).

I can't help myself at the denominator though. Am I obbliged to do Taylor series with center in 1?
And if I do, should I change the others too?
I was thinking even about applying log's properties, but that doesn't bring me too far.
Any suggestions?

2. ## Re: How could I solve this limit with Taylor series?

Originally Posted by dttah
Hello everyone.

I am not sure how to solve this with Taylor/Mc-Laurin series:

$\frac{\log(e^x-x-x^2)}{log\sin x-\log x}$

I have no problems at the numerator, I mean, I could do Mc-Laurin series for e^x so then I'd have a "1" which could help out in the following Mc-Laurin development:

log(1+x).

I can't help myself at the denominator though. Am I obliged to do Taylor series with center in 1?
And if I do, should I change the others too?
I was thinking even about applying log's properties, but that doesn't bring me too far.
Any suggestions?
For the denominator, follow your own suggestion and use properties of logs:

\begin{aligned}\log\sin x-\log x = \log\bigl(\tfrac{\sin x}{x}\bigr) &= \log\bigl(\tfrac{x-x^3/3! + x^5/5!-\ldots}x\bigr) \\ &= \log\bigl(1-(x^2/3!-x^4/5!+\ldots)\bigr) = \ldots .\end{aligned}

3. ## Re: How could I solve this limit with Taylor series?

Originally Posted by dttah
Hello everyone.

I am not sure how to solve this with Taylor/Mc-Laurin series:

$\frac{log(e^x-x-x^2)}{logsinx-logx}$

I have no problems at the numerator, I mean, I could do Mc-Laurin series for e^x so then I'd have a "1" which could help out in the following Mc-Laurin development:

log(1+x).

I can't help myself at the denominator though. Am I obbliged to do Taylor series with center in 1?
And if I do, should I change the others too?
I was thinking even about applying log's properties, but that doesn't bring me too far.
Any suggestions?
What value are you making x approach?

4. ## Re: How could I solve this limit with Taylor series?

Oh, righty, since I used the log's properties, I just could have done the senx/x taylor series. Thanks a lot!