# How do I use a power series to calculate a limit?

Oct 19, 2014

Here is a simple application of a power series in evaluating a limit.

${\lim}_{x \to 0} \frac{\sin x}{x}$

by replacing $\sin x$ by its Maclaurin series.

=lim_{x to 0}{x-x^3/{3!}+x^5/{5!}-x^7/{7!}+cdots}/{x}

by distributing the division to each term,

=lim_{x to 0}(1-x^2/{3!}+x^4/{5!}-x^6/{7!}+cdots)

by sending $x$ to zero,

$= 1 - 0 + 0 - 0 + \cdots$

since all but the first term are zero,

$= 1$

I hope that this was helpful.