find the limit as x -- 0 of (cosx - e^(-x^2/2))/x^4 ... using Taylor expansion ... !

Question

find the limit as x --> 0 of (cosx - e^(-x^2/2))/x^4 ... using Taylor expansion ... !

anonymous · Answer

UGLY!!
Firsly, do you know what a Taylor expansion is?

anonymous · Answer

$$f(x) = x^{-4}.(\cos x - e^{-x^2/2})$$
Set $$A = e^{-x^2/2}$$ so that dA/dx = -xA
$$f' = x^{-4}.(-\sin x + xA) -4x^{-5}.(\cos x - A)$$
rearranging
$$f' = -x^{-5}.[x\sin x +4\cos x - A(x^2+4)]$$
$$f''(x) = -x^{-5}.[-x\cos x +\sin x -4\sin x - 2xA + xA.(x^2+4)]$$
plus
$$-5x^{-6}.[x\sin x + 4\cos x - A(x^2+4)]$$
gather & rearrange, you get
$$f''(x) = -x^{-6}.[\cos x.(x^2-20) - 2x\sin x - A.(x^4 - 3x^2+20)]$$

anonymous · Answer

Taylor expansion:
$$f(x) = f(0) + x.f'(0) + x^2.f''(0)/2 + x^3.f'''(0)/3! + .....$$
Hope that helps - 10:30pm here & I've not started MY homework for the night yet!

anonymous · Answer

well thank you =]

mr.math · Answer

You can apply L'Hopital's rule few times since it's of $\frac{0}{0}$ form.

anonymous · Answer

I'm not sure how much help you need, or where you need it.
so... you get a few derivatives, enough to see the general pattern, which looks like x^{-n} as a factor with n getting bigger with each derivative.  In the other factor, you're getting cos or sin, with increasing powers of x multiplied to it, summed with A (the e term) multiplied by a polynomial of increasing powers of x.
L'Hopitals's rule is that if f(x) can be written in the form U/V, and the limit to whatever is undefined (eg 0/0 or infinity/infinity), then the same limit of f(x) is equal to U'/V', differentiating the numerator and denomiator totally separately.  Keep on applying L'Hopital's rule until you no longer get an undefined limit.

Now, to the Taylor expansion.  You can find the limit term by term because
limit(U+V) = lim U + lim V, but make sure you sub x=0 into those derivatives before forming up the taylor series.

anonymous · Answer

Oops - subbing 0 in makes the terms undefined, sorry.
the question seems to be asking not "what is the limit of the taylor series as x goes to 0", but FIND the taylor series (in terms of x) BY TAKING the limit as x goes to 0 of the UNSUBSTITUTED derivatives that form the taylor series.

If you need it explicity worked out for you - say so, if you're fine with what I've explained, well done - it's NOT an easy question to be given.