Taylor and the Limit

Taylor and the Limit

@ (Calculus)

In the limit of the solution, once used to use Taylor Formula + the form of the skin of the Connaught to think, may slowly have this is a good way of feeling it.

Consider, for example, a very common problem:

Limx→01−cosxcos2xcos3xx2=? \lim_{x\rightarrow 0}\frac{1-cosxcos2xcos3x}{x^2} =?

If skilled know equivalent infinitesimal: 1−cosx∼x22 1-cosx \sim \frac{x^2}{2}

But this source is still: cosx=1−x22!+x44!+...+ (−1) nx2n (2n)!+ ... cosx = 1-\frac{x^2}{2!} + \frac{x^4}{4!} + ... + ( -1) ^n\frac{x^{2n}}{( 2n)!} + ...

Narrow the angle of view, use 1−cosx∼x22 1-cosx \sim \frac{x^2}{2} to solve it.

First look at the molecule:

1−cosxcos2xcos3x=1−cosx+cosx−cosxcos2x+cosxcos2x−cosxcos2xcos3x=