\displaystyle
\lim_{x\rightarrow 0}\frac {\displaystyle\frac {x-vt} {\sqrt{1-(v/c)^2}}-x} {v}=
\lim_{x\rightarrow 0}\frac {\displaystyle\frac {x-vt} {\sqrt{1-(v/c)^2}}-(x-vt)-vt} {v}=
\lim_{x\rightarrow 0}\frac {\displaystyle(x-vt)\big(\frac {1} {\sqrt{1-(v/c)^2}}-1\big)} {v}-\lim_{v\rightarrow 0}\frac {vt} {v}=
\lim_{x\rightarrow 0}\frac {\displaystyle(x-vt)\frac {(v/c)^2} {\sqrt{1-(v/c)^2}\big(1+\sqrt{1-(v/c)^2}\big)}} {v}-t=-t.