4. The function f is defined by f(x) = e^xcos(x), x∈R.
a. Some derivatives for f(x) are given below. Determine the Maclaurin series for f(x) as far as the term in x^4.
| f(x) | e^xcos(x) |
| :----- | :--------------------- |
| f'(x) | e^xcos(x) - e^xsinx |
| f''(x) | -2e^xsin(x) |
| f'''(x)| -2e^xsin(x) - 2e^xcos(x)|
| f^(4)(x)| -4e^xcos(x) |
b. Hence, or otherwise, determine the exact value of lim (e^xcos(x)-x-1)/x^3
x→0