For any fixed $x>0$, find the value $$ \lim _{k \rightarrow \infty} \int_{0}^{x} \exp \left(-k t^{2} / 2\right) dt. $$
From the first sight, it seems that the function $f_k(t) = \exp(-k t^{2} / 2)$ will be uniformly convergent to some function $f_0$, which will guarantee the result of the suggested problem. However, this seems to be a little bit challenging to determine the exact value for $f_0$. Are there any ideas on the intuition behind determining such $f_0$ functions from the very beginning?!
Something to get you started: $f_k(t)\to 0$ uniformly for $t\in [0.1,x]$. Since $\int_0^x = \int_0^{0.1} + \int_{0.1}^x$, we have $$\limsup_{k\to\infty}\int_0^x f_k(t)\,dt \le 0.1.$$