Q:

A function that escapes to infinity with a finite input

I was wondering whether there exists a function that escapes to infinity with a finite input. For a specific example, how about $f(0)=0$ and as $x$ tends to $10$, $f(x)$ tends to infinity. The use of this would be to produce unimaginably large numbers with imaginable inputs.
If there are many such functions, I would prefer a formula which is short and snappy. If need be, magical operators such as an infinite sum or product would suffice. I understand that the tangent function has many poles (is that what you call an un-definition?), but they are sort of expensive to compute. I am not adept in calculus, but if you must... and at the very least I would like the function to be computable.
I am just a bit curious, is all. Can anyone help me? Perhaps the answer is obvious.

A:

Look at $\tan(x)$. https://www.wolframalpha.com/input/?i=tan%28x%29. As $x \to \frac\pi2$ then $f(x) \to \infty$. Also true for any integer multiple of $\frac\pi2$.
In general, functions of the form $f(x) = \frac{g(x)}{h(x)}$ will tend to infinity as $h(x)$ tends to 0, which doesn't necessarily require $x$ to tend to $\infty$. Disclaimer: $g(x)$ and $h(x)$ have to be well defined on the same interval, continuous and $h$ doesn't divide $g$ for all $x$ for this to be the case.
Here's an example that fits your criteria $f(0) = 0 $ and $f(x) \to \infty$ as $x \to 10$: $f(x) = \tan(\frac{\pi x}{20})$ If you want to construct a function $g$ where $g(x) \to \infty$ as $x \to k$ for some non-zero real number $k$, you can simply use $g(x)=\frac1{x-k}$.

