Question

Find out a function (fx) so that as x gets closer to 1, value of f(x) gets bigger?

Answer 1

Consider `f(x)` as a function of `x-1`.

When `x` gets closer to `1`, `x-1` becomes very small, so its reciprocal becomes very large in absolute value.

We can choose then any strictly increasing function `g(t)` and then let:

`f(x) = g(1/abs(x-1))`

or, to avoid discontinuities:

`f(x) = g(1/(x-1)^2)`

The simplest increasing function is:

`g(t) = t`

in which case we have:

`f(x) =1/(x-1)^2`

but we can also choose for instance:

`f(x) = e^(1/(x-1)^2)`