One way to disprove uniform convergence of a series of functions $f_n(x)$ in a domain D is to find the maximum of $f_n(x)$ at the domain and then to show that this maximum is bigger than an epsilon bigger than zero. This negates the definition of uniform convergence.
But sometimes there is no maximum of $f_n(x)$ in the domain. In this example : Show $\frac{\ln{(1+nx^2)}}{2n}$ is not uniform convergent on $[0,\infty)$ the user chooses $x= e^n$, then $f_n(e^n) \gt 1$ and if we choose $\epsilon = 1 $ then we are done. The same goes with an answer of a question I made : The uniform convergence of a series of functions depends on the domain
Now, my question is : How do I know the value of x? What is the trick? Unfortunately, the users did not go into detail on how they choose that particular x. And I cant understand it. I will appreciate some help.
Thanks!