Example of function with these features I am looking for an actual example of a convex function which has the following properties.

$f''(x)&gt;0$

$f'(0)\approx0$

$f'(1)\approx\infty$

It's easy to come up with an example which satisfies the first two:

$f(x)=x^{2}$

But the last one is not easy. 

Does anyone have any idea?
You didn't specify a domain, but if $[-1,1]$ is OK then $\sec(2x/\pi)$ has the required properties.
You do not specify a domain. But from $0$ to $π/2$, the function $\tan x$ satisfies all your requirements nicely.

The domain is an important part of any function. Are you working with complex numbers or just real ones? In what domain do you need this function to operate?

Also, within what range (assuming you're working on the real line) do you want the approximation $g(x)\approx0$ to fall?