Taylor's theorem

Exercise

1.: Show that the following MacLaurin expansion.
(a): $\displaystyle{\cos{x} = 1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \cdots + (-1)^{n}\frac{x^{2n}}{(2n)!} + \cdots,\ (-\infty < x < \infty)}$
(b): $\displaystyle{\log(1+x) = x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - \cdots + (-1)^{n-1}\frac{x^{n}}{n} + \cdots , \ (-1 < x \leq 1))}$
(c): $\displaystyle{(1+x)^{\alpha} = 1 + \frac{\alpha}{1!}x + \frac{\alpha(\alpha-1)}{2!}x^{2} + \cdots + \frac{\alpha(\alpha-1)\cdots(\alpha -n+1)}{n!}x^{n} + \cdots}$
where，
2.: Find the following limit using Landau's small o．
(a): $\displaystyle{\lim_{x \rightarrow 0}\frac{\log{(1+x)}}{x}}$
(b): $\displaystyle{\lim_{x \rightarrow 0}\frac{x - \sin{x}}{x^3}}$
(c): $\displaystyle{\lim_{x \rightarrow 0}\frac{e^{x} - 1 - x}{x^2}}$
(d): $\displaystyle{\lim_{x \rightarrow \infty}\frac{x^{\alpha}}{e^{x}}}$