Question

Решите пару предельчиков, пожалуйста:)

Answer 1

1)
$lim frac{( {2}^{x} - 1)lncosx}{ {x}^{2} times arcsin2x } = lim frac{ {2}^{x} - 1 }{arcsin2x} times lim frac{lncosx}{ {x}^{2} }$
1.
$lim frac{ {2}^{x} - 1}{arsin2x} = ( frac{0}{0}) = limfrac{ {2}^{x} ln2}{ frac{2}{ sqrt{1 - 4 {x}^{2} } } } = lim ({2}^{x - 1} ln2 sqrt{1 - 4 {x}^{2} } ) = {2}^{0 - 1} ln2 sqrt{1 - 4 times 0} = frac{ln2}{2}$
2.
$limfrac{lncosx}{ {x}^{2} } = ( frac{0}{0} ) = limfrac{ frac{1}{cosx} times ( - sinx)}{2x} = - frac{1}{2} lim frac{1}{cosx} times lim frac{sinx}{x} = - frac{1}{2} times frac{1}{cos0} times lim frac{cosx}{1} = - frac{1}{2} times 1 times cos0 = - frac{1}{2}$
$frac{ln(2)}{2} times ( - frac{1}{2} ) = - frac{ln(2)}{4}$
Ответ: - ln(2)/4.


2)
$lim(x - pi)lnsinx = lim frac{lnsinx}{ frac{1}{x - pi} } = lim frac{ frac{cosx}{sinx} }{ - frac{1}{ {(x - pi)}^{2} } } = - lim frac{ {(x - pi)}^{2} cosx}{sinx} = - limcosx times lim frac{ {(x - pi)}^{2} }{sinx} = - cospi times lim frac{ 2(x - pi) times ( - 1)}{cosx} = - 1 times ( - 1) times ( - 1) times limfrac{2(x - pi)}{cosx} = - frac{2(pi - pi)}{cospi} = - frac{2 times 0}{ - 1} = 0$
Ответ: 0