Use of Taylor's formula

Could anyone please help me with the following two problems?

1). f(x)=1/(1+x^3), determine f^100(0) (The 100th derivative at x=0.)

2). Find using Taylor's formula lim(x→0)(sinx-x)/(x(1-cosx))

Thanks a lot for looking and helping!

1) Use the geometric series.

Since 1/(1 - t) = Σ(n = 0 to ∞) t^n, letting t = -x^3 yields
1/(1 + x^3) = Σ(n = 0 to ∞) (-1)^n x^(3n).

Since the coefficient of x^100 is 0 (as the only nonzero coefficients are for exponents which are multiples of 3), and the Macluarin coefficient for x^100 is f^(100)(0) / 100!, we see that
f^(100)(0) / 100! = 0 ==> f^(100)(0) = 0.
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2) Use the familiar series for sine and cosine.

lim(x→0) (sin x - x) / [x (1 - cos x)]
= lim(x→0) [(x - x^3/3! + x^5/5! + ...) - x] / [x (1 - (1 - x^2/2! + x^4/4! - ...)]
= lim(x→0) (-x^3/3! + x^5/5! + ...) / [x (x^2/2! - x^4/4! + ...)]
= lim(x→0) x^3 (-1/3! + x^2/5! + ...) / [x^3 (1/2! - x^2/4! + ...)]
= lim(x→0) (-1/3! + x^2/5! + ...) / (1/2! - x^2/4! + ...)
= (-1/3! + 0) / (1/2! - 0)
= -1/3.

I hope this helps!