Evaluate the limit.
![\[ \lim_{x \to 0} \frac{\cosh x - \cos x}{x^2}. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-6ec51ef74f0959e80454663a58b2c4eb_l3.png "Rendered by QuickLaTeX.com")
We use the definition of 
in terms of the exponential:
![\[ \cosh x = \frac{e^x + e^{-x}}{2}, \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-49b7f19fc33c04e8939b9ad6cdc81434_l3.png "Rendered by QuickLaTeX.com")
and the expansions (page 287 of Apostol) of 
and 
as 
:
Putting these together we evaluate the limit:
![\begin{align*} \lim_{x \to 0} \frac{\cosh x - \cos x}{x^2} &= \lim_{x \to 0} \frac{e^x + e^{-x} - 2\cos x}{2x^2} \\[9pt] &= \lim_{x \to 0} \frac{1 + x + \frac{x^2}{2} + \frac{x^3}{6} + o(x^3) + 1 - x + \frac{x^2}{2} - \frac{x^3}{6} + o(x^3) - 2 \cos x}{2x^2} \\[9pt] &= \lim_{x \to 0} \frac{2 + x^2 + o(x^3) - 2\left(1 - \frac{x^2}{2} + o(x^3) \right)}{2x^2} \\[9pt] &= \lim_{x \to 0} \frac{2x^2 + o(x^3)}{2x^2} \\[9pt] &= \lim_{x \to 0} \left( 1 + \frac{o(x^3)}{x^2} \right) \\[9pt] &= 1. \end{align*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-38a4f7507112cad6301ed2765d17cd1e_l3.png "Rendered by QuickLaTeX.com")
Evaluate the limit.
We use the definition of in terms of the exponential:
and the expansions (page 287 of Apostol) of and as :
Putting these together we evaluate the limit:
A bit confused about why we have 2cosx, when originally we start if with just cosx.
keep in mind that cosh(x) = 1/2 * (e^x + e^(-x)), he made a common denominator for cosh and cos