real analysis - Show that $e^{varepsilon |x|^{varepsilon}}$ grows faster than $sum_{k=0}^{infty} {|x|^{2k}}/{(k!)^2}$

I am wondering whether we have for $$f(x):=\sum_{k=0}^{\infty} \frac{|x|^{2k}}{(k!)^2} $$ that

$$\lim_{x \rightarrow \infty} \frac{e^{\varepsilon |x|^{\varepsilon}}}{f(x)} = \infty$$ for any $\varepsilon>0$?

I assume that this is true as factorials should somehow outgrow powers, but I do not see how to show this rigorously?

Does anybody have an idea?