Limit

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Jan 27th 2008, 06:09 PM
Adonis

Limit

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

Thank you
Jan 27th 2008, 07:50 PM
mr fantastic

Quote:

Originally Posted by Adonis

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

Thank you

Substitute the Maclaurin Series for cos x, simplify and take the limit:

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

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

As x --> 0 the limiting value is obvious .....
Jan 27th 2008, 07:54 PM
ThePerfectHacker

Quote:

Originally Posted by Adonis

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

$\frac{1-\cos x}{x^2} \cdot \frac{1+\cos x}{1+\cos x} = \frac{\sin^2 x}{x^2} \cdot \frac{1}{1+\cos x}$ .

Now use the fact that $\lim_{x\to 0}\frac{\sin x}{x} = 1$ .

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Using infinite series is like shooting a dead fly with a Grand Piano.
Jan 27th 2008, 08:26 PM
mr fantastic

Quote:

Originally Posted by ThePerfectHacker

[tex][snip]
Using infinite series is like shooting a dead fly with a Grand Piano.

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 AM
curvature

Quote:

Originally Posted by Adonis

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

Thank you

Use the equality:1-cosx=2sin(x/2)^2.
Jan 28th 2008, 04:47 AM
mr fantastic

Quote:

Originally Posted by curvature

Use the equality:1-cosx=2sin(x/2)^2.

Now thats what I call a rabbit (Rofl)