[latex] \lim_{x \to\ 0} e^4^x-e^-^2^x/2arctgx-sinx[/latex]

[latex] \lim_{x \to 0} e^{4x} - e^{ \frac{-2x}{2arctg(x) - sin(x)} } \lim_{x \to 0}e^{4x} - \lim_{x \to 0}e^{ \frac{-2x}{2arctg(x) - sin(x)} } 1 - \lim_{x \to 0}e^{ \frac{-2x}{2arctg(x) - sin(x)} } 1 - e^{\lim_{x \to 0} \frac{-2x}{2arctg(x) - sin(x)} } 1 - e^{ \frac{\lim_{x \to 0}-2x}{\lim_{x \to 0}(2arctg(x) - sin(x))} } 1 - e^{ -2\frac{-x}{\lim_{x \to 0}(2arctg(x) - \lim_{x \to 0sin(x))} } 1 - e^{ -2\frac{-x}{\lim_{x \to 0}(2arctg(x) - x} } 1 - e^{ -2\frac{-x}{0 - x} 1 - e^{ -2} [/latex] [latex] \lim_{x \to 0} e^{4x} - e^{ \frac{-2x}{2arctg(x) - sin(x)} } = 1 -e^{-2}[/latex]