Question

lim 1-cos^2 5x/3x^2,где x->0
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Answer 1

$lim _{xto :0}left(frac{1-cos ^2left(5xright)}{3x^2}right)=frac{1}{3}cdot lim :_{xto :0}left(frac{1-cos ^2left(5xright)}{x^2}right)= \ \ =frac{1}{3}cdot lim :_{xto :0}left(frac{5sin left(10xright)}{2x}right)=frac{1}{3}cdot lim :_{xto :0}left(frac{50cos left(10xright)}{2}right)= \ \ =frac{1}{3}cdot lim :_{xto :0}left(25cos left(10xright)right)=frac{1}{3}cdot :25cos left(10cdot :0right)=frac{25}{3} \ \ OTVET: frac{25}{3}$

Answer 2

$limlimits _{x to 0} frac{1-cos^25x}{3x^2} = limlimits _{x to 0} frac{sin^25x}{3x^2}= limlimits _{x to 0}Big (underbrace {frac{sin5x}{5x}}_{1}cdot underbrace {frac{sin5x}{5x}}_{1}cdot frac{25x^2}{3x^2}Big )=\\= limlimits _{x to 0} frac{25x^2}{3x^2} = frac{25}{3}$