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