Question

LIM=(TGX-SINX)/(3X^2) ПРИ X=0

Answer 1

$L=limlimits_{xto 0}dfrac{tan x-sin x}{3x^2}\\L=dfrac{1}{3}limlimits_{xto 0}dfrac{(1-cos x)sin x}{x^2cos x}\\L=dfrac{1}{3}limlimits_{xto 0}dfrac{1-cos x}{x}cdotdfrac{sin x}{x}cdotdfrac{1}{cos x}\\L=dfrac{1}{3}limlimits_{xto 0}dfrac{(1-cos x)(1+cos x)}{x(1+cos x)}cdot1cdot1\\L=dfrac{1}{3}limlimits_{xto 0}dfrac{1-cos^2 x}{x(1+cos x)}$

$L = dfrac{1}{3} lim limits_{x to 0} dfrac{sin ^ 2 x}{x (1+ cos x)} \\L = dfrac{1}{3} lim limits_{x to 0} dfrac{sin x}{1+ cos x} cdot dfrac{sin x}{x} \\L = dfrac{1}{3} lim limits_{x to 0} dfrac{sin x}{1+ cos x} cdot 1 \\L = dfrac{1}{3} lim limits_{x to 0} dfrac{sin x}{1+ cos x} \\L = dfrac{1}{3} cdot dfrac{0}{2} \\boxed{L = 0}$