Final answer:
To evaluate the limit lim x → 0 (1 - cos(3x))/(1 + 3x - e^3x) using series, substitute the Taylor series expansions of cosine and exponential functions, cancel out the x term in the denominator, and substitute x = 0 to find the limit value of 1/2.
Step-by-step explanation:
To evaluate the limit lim x → 0 (1 - cos(3x))/(1 + 3x - e^3x) using series, we can use the Taylor series expansion of cosine and exponential functions. We know that the Taylor series expansion of cos(x) is 1 - (x^2)/2! + (x^4)/4! - ..., and the Taylor series expansion of e^x is 1 + x + (x^2)/2! + ...
Using these expansions, we can substitute them into our original limit expression and cancel out the x term in the denominator. This leaves us with the series expansion of (1 - cos(3x))/(1 - x). Now we can substitute x = 0 into this expression to evaluate the limit, which gives us the value of 1/2.