lim_{x-0} [x*(e^x - 1) + 2*(cos x - 1)] / [x*(1 - cos x)] is __________.

Question

lim_{x->0} [x*(e^x - 1) + 2*(cos x - 1)] / [x*(1 - cos x)] is __________.

Accepted Answer

1 Both numerator and denominator vanish as x approaches 0, giving the 0/0 indeterminate form. Replacing each function by the first few terms of its Maclaurin series reveals the leading-order behavior: the numerator is O(x^3) and the denominator is also O(x^3), so their ratio has a finite non-zero limit. Dividing out x^3 and taking x to 0 gives the limit.