[latex] \lim_{x \to 0} \frac{2x*sinx}{secx-1} =[/latex]

[latex]lim_{x\to0}\frac{2x*sinx}{secx}=lim_{x\to0}\frac{2x*sinx}{\frac{1-cosx}{cosx}}=\\=[1-cosx=2sin^2\frac{x}{2};sinx=2sin\frac{x}{2}cos\frac{x}{2}]=\\=lim_{x\to0}\frac{2x*2sin\frac{x}{2}cos\frac{x}{2}*cosx}{2sin^2\frac{x}{2}}=lim_{x\to0}\frac{4*\frac{x}{2}*cos\frac{x}{2}*cosx}{sin\frac{x}{2}}=\\=[lim_{x\to0}\frac{\frac{x}{2}}{sin\frac{x}{2}}=lim_{x\to0}\frac{1}{\frac{sin\frac{x}{2}}{\frac{x}{2}}}=\frac{1}{1}=1]=\\=lim_{x\to0}(4*1*cos\frac{x}{2}*cosx)=4*cos0*cos0=4*1*1=4[/latex]