exponential function - How to show limit containing exp and cos series inside?

I've got some limit to show.

$$\lim_{x \rightarrow \infty} \frac{\sum_{n=0}^{\infty}( \frac{x^n}{n!})-1}{1-\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n}}{(2n)!}}$$ What is equivalent to $$\lim_{x \rightarrow \infty} \frac{exp(x)-1}{1-cos(x)}$$

I tried to split it into "even" part and "odd" part, i mean first calculate for $n=0,2,4,6\dots$ and then for $n=1,3,5,7\dots$ but it all got messy and not really led me to any solution. This is my first this kind of task i have to solve, so i don't know any good tricks/ideas i can use here...

I'd appreciate some help! thanks

EDIT: some people state that this limit doesn't exist. But just being curious, what if we want to do series of out it? Seems like it will diverge, right? Why then wolframalpha gives answer = INF to this limit?

Answer

Since $2\geq 1-cos(x) \geq 0$ for all $x$, we have

$$\frac{e^x - 1}{1 - cos(x)} > \frac{e^x - 1}{2}$$

This value becomes arbitrarly large as $x$ becomes large. This means the limit, if any, is $\infty$ (this is of course a generalized limit, no standard limit exists)

Note, however, that even this generalized limit does not actually exist as the function is not defined for all $x$ of the type $x=2\pi k$ for $k\in\mathbb{N}$.