Chapter 11 - Infinite Sequences and Series - 11.10 Taylor and Maclaurin Series - 11.10 Exercises - Page 812: 62

$-1$

$cosx=1-\frac{x^{2}}{2!}+\frac{x^{4}}{4!}-....$ and $e^{x}=1+x+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}+...$ Plug into the limit to get $\lim\limits_{x \to 0}\frac{1-cosx}{1+x-e^{x}}=\lim\limits_{x \to 0}\frac{1-1-\frac{x^{2}}{2!}+\frac{x^{4}}{4!}-....}{1+x-1+x+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}+...}$ $=\frac{1/2}{-1/2}$ $=-1$