Question

Замечательные пределы:
$lim_{n to 0} frac{sin frac{x}{2} }{sin3x} lim_{n to 0} frac{1-cos2x}{x^2} lim_{n to 0} frac{sin^{2}x }{sin2x} lim_{n to 0} (frac{2x+1}{2x-3})^{x}$

Answer 1

$lim_{x to 0} frac{sin frac{x}{2} }{sin 3x} = lim_{x to 0} frac{3xcdot frac{x}{2} cdotsin frac{x}{2} }{3xcdot frac{x}{2} cdotsin3x} = lim_{x to 0} frac{ frac{x}{2} }{3x} = frac{1}{6}$

$lim_{x to 0} frac{1-cos2x}{x^2} = frac{1-1+2sin^2x}{x^2}= lim_{x to 0} frac{2sin^2x}{x^2} =2$

$lim_{x to 0} frac{sin^2x}{sin2x}= lim_{x to 0} frac{sin^2x}{2sin xcos x} = lim_{x to 0} frac{sin x}{2cos x}= lim_{x to 0} frac{xsin x}{2xcos x} =0$

$lim_{x to 0} ( frac{2x+1}{2x-3} )^x= lim_{x to 0} ( frac{2x-3+4}{2x-3} )^x= lim_{x to 0} (1+ frac{4}{2x-3} )^x=\ = lim_{x to 0} (1+ frac{4}{2x-3} )^{xcdot frac{4}{2x-3} cdot frac{2x-3}{4} }=e^{ lim_{x to 0} frac{4x}{2x-3} }=e^0=1$