By definition of tangent,
[tex]\tan x=\dfrac{\sin x}{\cos x}[/tex]
So
[tex]\displaystyle\lim_{x\to0}(\tan2x-\sin2x)=\lim_{x\to0}\sin2x\left(\frac1{\cos2x}-1\right)[/tex]
[tex]\displaystyle=\left(\lim_{x\to0}\sin2x\right)\left(\lim_{x\to0}\left(\frac1{\cos2x}-1\right)\right)=0[/tex]
