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

[latex]lim_{xo0}frac{2x*sinx}{secx}=lim_{xo0}frac{2x*sinx}{frac{1-cosx}{cosx}}=\=[1-cosx=2sin^2frac{x}{2};sinx=2sinfrac{x}{2}cosfrac{x}{2}]=\=lim_{xo0}frac{2x*2sinfrac{x}{2}cosfrac{x}{2}*cosx}{2sin^2frac{x}{2}}=lim_{xo0}frac{4*frac{x}{2}*cosfrac{x}{2}*cosx}{sinfrac{x}{2}}=\=[lim_{xo0}frac{frac{x}{2}}{sinfrac{x}{2}}=lim_{xo0}frac{1}{frac{sinfrac{x}{2}}{frac{x}{2}}}=frac{1}{1}=1]=\=lim_{xo0}(4*1*cosfrac{x}{2}*cosx)=4*cos0*cos0=4*1*1=4[/latex]