Past exam of the mathematical methods II course of the University of Oxford / 2013 / 2

Put $h=x-\pi /2$. Periodicity gives $\tan (2x)=\tan (\pi +2h)=\tan (2h)$. The local Taylor expansion $\tan (2h)=2h+O(h^3)$ therefore yieldsAlthough $\tan x$ is singular at $\pi /2$, the numerator here is $\tan (2x)$, which is smooth and zero there.

Continuity at $x=2$ requires both one-sided limits to equal the specified value $3$. The right branch gives $a+2b=3$ and the left branch gives $b-4a=3$. Solving these two linear equations givesSubstitution gives $-1/3+10/3=3$ on the right and $5/3+4/3=3$ on the left. These conditions enforce continuity, without requiring the two one-sided derivatives to agree.