Question

Нужно подробное решение правилом Лопиталя.

Answer 1

$displaystyleLargelim_{xto0}{x-sin{x}over x-tg{x}}=lim_{xto0} {1-cos{x}over 1-{1overcos^2{x}}}=lim_{xto0} {(1-cos{x})cdotcos^2{x}over cos^2{x}-1}=lim_{xto0}{cos^2{x}-cos^3{x}overcos^2{x}-1}=lim_{xto0}{-2cos{x}sin{x}+3cos^2{x}sin{x}over-2cos{x}sin{x}}=lim_{xto0}{2cos{x}-3cos^2{x}over2cos{x}}={2-3over2}=-{1over2}$