Im sorry but I donīt understand much of this? Isnīt the point when calculating with lim that if then you put x=0 in the expression. When i do that i get only zeros.
And what happens to the x and 1 in x(cosx-1)? Same with the -x in sinx - x??
And how do you get that the cosx (mclaurin...) expression is approximately and the same with the sin expression?
Hope you have time for at least a brief explanation.
I have done some editing to Mathstud28's reply, but you should have realised (and in fact done) these things yourself:
Now divide numerator and denominator by :
Now "put x = 0 in the expression" as you say it: