Using Series to Evaluate Limits and Integrals

A limit like (sin(x) - x)/x³ looks like 0/0 no matter how you stare at it, because sin(x) and x agree so closely near 0 that the numerator hides its own behavior. Swap sin(x) for its Maclaurin series x - x³/6 + x⁵/120 - ..., and the x cancels instantly, leaving -x³/6 + x⁵/120 - ... over x^3. Divide through and every surviving term has a positive power of x, so as x → 0 they vanish and the answer falls out: -1/6. The series turns a mystery cancellation into ordinary algebra.

Integrals work the same way when no elementary antiderivative exists, like the integral of sin(x²). You cannot write down a closed-form antiderivative for that function, but you can substitute x² into sin's series, getting x² - x⁶/6 + x¹⁰/120 - ..., and integrate term by term since each term is just a power of x. That produces x³/3 - x⁷/42 + x¹¹/1320 - ..., a series you can evaluate at the bounds and truncate once the terms get tiny. This is the same term-by-term logic used for power series representations and radius of convergence earlier in the unit, now put to work.

This method is a close cousin of L'Hospital's rule. Both handle 0/0 limits, but L'Hospital differentiates repeatedly and can require several rounds for higher-order vanishing, while a series expansion gets all the cancellation at once by exposing every term's power of x simultaneously.