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Консультация № 111427
28.11.2007, 20:15
0.00 руб.
0
2
2
Найти предлы, использую правило Лопиталя:
а). lim(x(lnx)^3) x стремится к 0
б). lim(x^2(1-e^(1/x))) x стремится к бесконечности
в). lim((sin x + cos x)/(x+p/4)) x стремится к p/4 (p-пи)
г). lim((ln(1+x)-ln2)/(√(1+2x)-3^(x-1/2))) x стремится к 1
д). lim(x+√(x))^1/x x стремится к бесконечности
e). lim(ln ln(x^2-3)/ln(x-2)) x стремится к 2
ж). lim (sinx + √(x))^x x стремится к 0
з). lim arcsin x/sin s x стремится к 0
а). lim(x(lnx)^3) x стремится к 0
б). lim(x^2(1-e^(1/x))) x стремится к бесконечности
в). lim((sin x + cos x)/(x+p/4)) x стремится к p/4 (p-пи)
г). lim((ln(1+x)-ln2)/(√(1+2x)-3^(x-1/2))) x стремится к 1
д). lim(x+√(x))^1/x x стремится к бесконечности
e). lim(ln ln(x^2-3)/ln(x-2)) x стремится к 2
ж). lim (sinx + √(x))^x x стремится к 0
з). lim arcsin x/sin s x стремится к 0
Обсуждение
Неизвестный
29.11.2007, 23:42
общий
это ответ
Здравствуйте, Kamelia26!
а)
lim<sub>x→0</sub>x*ln³x = lim<sub>x→0</sub>ln³x/(1/x) =
= {применим правило Лопиталя} =
= lim<sub>x→0</sub>(ln³x)‘/(1/x)‘ = lim<sub>x→0</sub>(3ln²x*1/x)/(-1/x²) = lim<sub>x→0</sub>(-3ln²x)/(1/x) =
= {применим правило Лопиталя во второй раз} =
= lim<sub>x→0</sub>(-3ln²x)‘/(1/x)‘ = lim<sub>x→0</sub>(-6ln(x)*1/x)/(-1/x²) = lim<sub>x→0</sub>(6ln(x))/(1/x) =
= {в третий раз воспользуемся правилом Лопиталя} =
= lim<sub>x→0</sub>(6ln(x))‘/(1/x)‘ = lim<sub>x→0</sub>(6/x)/(-1/x²) = lim<sub>x→0</sub>(-6x) = 0.
а)
lim<sub>x→0</sub>x*ln³x = lim<sub>x→0</sub>ln³x/(1/x) =
= {применим правило Лопиталя} =
= lim<sub>x→0</sub>(ln³x)‘/(1/x)‘ = lim<sub>x→0</sub>(3ln²x*1/x)/(-1/x²) = lim<sub>x→0</sub>(-3ln²x)/(1/x) =
= {применим правило Лопиталя во второй раз} =
= lim<sub>x→0</sub>(-3ln²x)‘/(1/x)‘ = lim<sub>x→0</sub>(-6ln(x)*1/x)/(-1/x²) = lim<sub>x→0</sub>(6ln(x))/(1/x) =
= {в третий раз воспользуемся правилом Лопиталя} =
= lim<sub>x→0</sub>(6ln(x))‘/(1/x)‘ = lim<sub>x→0</sub>(6/x)/(-1/x²) = lim<sub>x→0</sub>(-6x) = 0.
Неизвестный
30.11.2007, 00:40
общий
это ответ
Здравствуйте, Kamelia26!
e). lim<sub>x→2</sub>ln(ln(x<sup>2</sup>-3))/ln(x-2) = {применяем правило Лопиталя} = lim<sub>x→2</sub> (2x/((x<sup>2</sup>-3)*ln(x<sup>2</sup>-3)))/1/(x-2) = lim<sub>x→2</sub> 2x*(x-2)/((x<sup>2</sup>-3)*ln(x<sup>2</sup>-3)) =
= {применяем правило Лопиталя повторно} = lim<sub>x→2</sub> (4x - 4)/(2x + 2x*ln(x<sup>2</sup>-3)) = (4*2-4)/(2*2 + 2*0) = 4/4 = 1.
Good Luck!!!
e). lim<sub>x→2</sub>ln(ln(x<sup>2</sup>-3))/ln(x-2) = {применяем правило Лопиталя} = lim<sub>x→2</sub> (2x/((x<sup>2</sup>-3)*ln(x<sup>2</sup>-3)))/1/(x-2) = lim<sub>x→2</sub> 2x*(x-2)/((x<sup>2</sup>-3)*ln(x<sup>2</sup>-3)) =
= {применяем правило Лопиталя повторно} = lim<sub>x→2</sub> (4x - 4)/(2x + 2x*ln(x<sup>2</sup>-3)) = (4*2-4)/(2*2 + 2*0) = 4/4 = 1.
Good Luck!!!
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