what is $\lim_{x\to\infty}\frac{\cosh x}{x!}?$
I got 0 for this, as it seems $x!$ grows at a much faster rate than $\cosh x$ ?
what is $\lim_{x\to\infty}\frac{\cosh x}{x!}?$
I got 0 for this, as it seems $x!$ grows at a much faster rate than $\cosh x$ ?
HINT: By definition $\cosh n=\frac12(e^n+e^{-n})$, so
$0<\frac{\cosh n}{n!}=\frac{e^n+e^{-n}}{2n!}<\frac{e^n}{n!}\;;$
can you show that $\lim_{n\to\infty}\frac{e^n}{n!}=0\;?$
Note that $\frac{e^n}{n!}=\frac{\overbrace{e\cdot e\cdot e\cdot\ldots\cdot e}^{n\text{ factors}}}{1\cdot2\cdot3\cdot\ldots\cdot n}\;.$