$\displaystyle \lim_{x\to 0}\frac{1-cos\;x}{x^2}$ without l'Hopital Rule.

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- Jan 27th 2008, 06:09 PMAdonisLimit
$\displaystyle \lim_{x\to 0}\frac{1-cos\;x}{x^2}$ without l'Hopital Rule.

Thank you - Jan 27th 2008, 07:50 PMmr fantastic
Substitute the Maclaurin Series for cos x, simplify and take the limit:

$\displaystyle \cos x = 1 - \frac{x^2}{2} + \frac{x^4}{24} - ......$

Therefore you have $\displaystyle \frac{1 - 1 + \frac{x^2}{2} - \frac{x^4}{24} + .....}{x^2} = \frac{1}{2} - \frac{x^2}{24} + .....$.

As x --> 0 the limiting value is obvious ..... - Jan 27th 2008, 07:54 PMThePerfectHacker
- Jan 27th 2008, 08:26 PMmr fantastic
True.

But I've found most students can be taught to kill even a**live**fly using a Grand Piano. As opposed to teaching them how to pull a rabbit out of a hat (even when it's a fairly ordinary rabbit, as in the present case) .....

By the way, I don't mean to be critical of the rabbit - I love magic - it's just that most students struggle to learn it. Even when they do, they usually pull out a dead fish .....

My view, all comment welcome. - Jan 28th 2008, 03:34 AMcurvature
- Jan 28th 2008, 04:47 AMmr fantastic