Does Taylor Series accurately represent limits in calculus?

[ironman]

Homework Statement

[/B]
lim x -> 0

2. Homework Equations
Taylor series for sin cos e and ln ()

The Attempt at a Solution

I tried expanding the sine to 3-degree, and everything else 2-degree. I ended up with this:

Now the problem is that WolframAlpha says it should be -6/25. Now if only that -2 were +2...

 

[ironman]

I got the answer! the first 1/2 (expansion from e^3x) is supposed to be 4 1/2 not 1/2, because you get (3x)^2 / 2! not (x)^2/2!