12. Consider the hyperbolic sine function defined by sinh(x) := 7 (ex -e-*) (a) Showing all your working, find the first five derivatives of f (x) := sinh(x) at x = 0 (i.e., find f' (0), f"(0), f(3) (0), f(+) (0), f(5)(0)) and hence write down the first few terms, up to and including x5, of the Taylor series for sinh(x ) about x = 0. (b) Use the Taylor series for sinh(x) from part (a) to do the following. i. Estimate the definite integral - x sinh(x) dx ii. Find lim sinh(x) x-+0 X Also verify that the limit value you obtain using the Taylor series for sinh(x) is the same as what you get using L'Hopital's Rule