Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics II / 5 / c / i / Solution

At $x=0$, $f_n(x)=0$. For fixed $x>0$,Thus the pointwise limit is zero everywhere on $[0,1]$. However, with $t=nx$, the inequality $1+t^2\ge 2t$ gives $f_n(x)\le 1/2$, and equality occurs at $x=1/n\in [0,1]$. Henceso there is no uniform convergence. The moving evaluation point $1/n$ captures a peak missed by each fixed positive input.