e^x and sinx approximated by cubic functions when x is small

[Idealistic]

Given the lim[sub:ds3jduor]x-->0[/sub:ds3jduor] (e[sup:ds3jduor]x[/sup:ds3jduor] - 1 - x - (1/2)x[sup:ds3jduor]2[/sup:ds3jduor])/x[sup:ds3jduor]3[/sup:ds3jduor] = 1/6, and

limx[sub:ds3jduor]x-->0[/sub:ds3jduor] (sinx - x)/x[sup:ds3jduor]3[/sup:ds3jduor] = -1/6, how can I show or assume that e[sup:ds3jduor]x[/sup:ds3jduor] and sinx can be approximated by cubic functions when x is small? I think I'm missing something

.

 

[galactus]

By cubic, the best thing I can think of is the Taylor series for e^x and sin(x).

Using:

\(\displaystyle e^{x}=\frac{x^{3}}{6}+\frac{x^{2}}{2}+x+1\)


\(\displaystyle sin(x)=\frac{-x^{3}}{6}+x\)

but ignoring the other terms.


For e^x, let's try x=.000001. Which is pretty small.

Using the series, we get 1.000001

and \(\displaystyle e^{.000001}=1.000001\)