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# Задачи за граници на тригонометриски функции

Module by: Beti Andonovic. E-mail the author

Summary: Solved exercises on limits of trigonometric functions Решени задачи од лимеси на тригонометриски функции

Тригонометриски лимеси

1. limx0sin3xx=limx0cos2xsin3xx=limx0cosxsin3x3x3xxx=limx0sin3xx=limx0cos2xsin3xx=limx0cosxsin3x3x3xxx= size 12{ {"lim"} cSub { size 8{x rightarrow 0} } { {"sin"3x} over {x} } = {"lim"} cSub { size 8{x rightarrow 0} } { {"cos"2x"sin"3x} over {x} } = {"lim"} cSub { size 8{x rightarrow 0} } { {"cos"x { {"sin"3x} over {3x} } 3x} over { { {x} over {x} } } } ={}} {}

= lim x 0 3x cos x sin 3x 3x 1 = lim x 0 3x cos x 1 = lim x 0 3 cos 0 1 = 3 1 = 3 = lim x 0 3x cos x sin 3x 3x 1 = lim x 0 3x cos x 1 = lim x 0 3 cos 0 1 = 3 1 = 3 size 12{ {}= {"lim"} cSub { size 8{x rightarrow 0} } { {3x"cos"x { {"sin"3x} over {3x} } } over {1} } = {"lim"} cSub { size 8{x rightarrow 0} } { {3x"cos"x} over {1} } = {"lim"} cSub { size 8{x rightarrow 0} } { {3"cos"0} over {1} } = { {3} over {1} } =3} {}
(1)

2. limx0sinαxsinβx=limx0αsinαxαxβsinβxβx=αβsin0sin0=0limx0sinαxsinβx=limx0αsinαxαxβsinβxβx=αβsin0sin0=0 size 12{ {"lim"} cSub { size 8{x rightarrow 0} } { {"sin"αx} over {"sin"βx} } = {"lim"} cSub { size 8{x rightarrow 0} } { {α { {"sin"αx} over {αx} } } over {β { {"sin"βx} over {βx} } } } = { {α} over {β} } cdot { {"sin"0} over {"sin"0} } =0} {}

3. limx0tgkxx=limx0sinkxcoskxx=limx0sinkxxcoskxkk=limx0(ksinkxkx1coskx)=klimx0tgkxx=limx0sinkxcoskxx=limx0sinkxxcoskxkk=limx0(ksinkxkx1coskx)=k size 12{ {"lim"} cSub { size 8{x rightarrow 0} } { {"tg" ital "kx"} over {x} } = {"lim"} cSub { size 8{x rightarrow 0} } { { { {"sin" ital "kx"} over {"cos" ital "kx"} } } over {x} } = {"lim"} cSub { size 8{x rightarrow 0} } { {"sin" ital "kx"} over {x"cos" ital "kx"} } cdot { {k} over {k} } = {"lim"} cSub { size 8{x rightarrow 0} } $${ {k"sin" ital "kx"} over { ital "kx"} } { {1} over {"cos" ital "kx"} }$$ =k} {}

4. limx0tan2xsin5x=limx0sin2xcos2xsin5x=limx025xsin2x25xcos2xsin5x=limx0tan2xsin5x=limx0sin2xcos2xsin5x=limx025xsin2x25xcos2xsin5x= size 12{ {"lim"} cSub { size 8{x rightarrow 0} } { {"tan"2x} over {"sin"5x} } = {"lim"} cSub { size 8{x rightarrow 0} } { {"sin"2x} over {"cos"2x"sin"5x} } = {"lim"} cSub { size 8{x rightarrow 0} } { {2 cdot 5x"sin"2x} over {2 cdot 5x"cos"2x"sin"5x} } ={}} {}

= lim x 0 ( 2 5 cos 2x sin 2x 2x sin 5x 5x ) = 2 5 1 1 = 2 5 = lim x 0 ( 2 5 cos 2x sin 2x 2x sin 5x 5x ) = 2 5 1 1 = 2 5 size 12{ {}= {"lim"} cSub { size 8{x rightarrow 0} } $${ {2} over {5"cos"2x} } { { { {"sin"2x} over {2x} } } over { { {"sin"5x} over {5x} } } }$$ = { {2} over {5} } cdot { {1} over {1} } = { {2} over {5} } } {}
(2)

5. limx0sin(xn)(sinx)m=limx0sin(xn)sinmx=limx0sin(xn)xnsinmxxn=limx0sin(xn)(sinx)m=limx0sin(xn)sinmx=limx0sin(xn)xnsinmxxn= size 12{ {"lim"} cSub { size 8{x rightarrow 0} } { {"sin" $$x rSup { size 8{n} }$$ } over { $$"sin"x$$ rSup { size 8{m} } } } = {"lim"} cSub { size 8{x rightarrow 0} } { {"sin" $$x rSup { size 8{n} }$$ } over {"sin" rSup { size 8{m} } x} } = {"lim"} cSub { size 8{x rightarrow 0} } { { { {"sin" $$x rSup { size 8{n} }$$ } over {x rSup { size 8{n} } } } } over { { {"sin" rSup { size 8{m} } x} over {x rSup { size 8{n} } } } } } ={}} {}

= lim x 0 1 sin m x x n + m m = lim x 0 1 sin m x x m x n m = 1 1 x n m = { 0,n > m 1,n = m ,n < m = lim x 0 1 sin m x x n + m m = lim x 0 1 sin m x x m x n m = 1 1 x n m = { 0,n > m 1,n = m ,n < m alignl { stack { size 12{ {}= {"lim"} cSub { size 8{x rightarrow 0} } { {1} over { { {"sin" rSup { size 8{m} } x} over {x rSup { size 8{n+m - m} } } } } } = {"lim"} cSub { size 8{x rightarrow 0} } { {1} over { { {"sin" rSup { size 8{m} } x} over {x rSup { size 8{m} } cdot x rSup { size 8{n - m} } } } } } = { {1} over {1} } x rSup { size 8{n - m} } = left lbrace matrix { "0,n ">" m " {} ## "1,n"=m {} ## infinity ",n"<m } right none } {} # {} } } {}
(3)

6. limx01cosxx2=limx02sin2x2x2=2limx0(sinx2x)2=2limx0(sinx22x2)2=limx01cosxx2=limx02sin2x2x2=2limx0(sinx2x)2=2limx0(sinx22x2)2= size 12{ {"lim"} cSub { size 8{x rightarrow 0} } { {1 - "cos"x} over {x rSup { size 8{2} } } } = {"lim"} cSub { size 8{x rightarrow 0} } { {2"sin" rSup { size 8{2} } { {x} over {2} } } over {x rSup { size 8{2} } } } =2 {"lim"} cSub { size 8{x rightarrow 0} } $${ {"sin" { {x} over {2} } } over {x} }$$ rSup { size 8{2} } =2 {"lim"} cSub { size 8{x rightarrow 0} } $${ {"sin" { {x} over {2} } } over {2 cdot { {x} over {2} } } }$$ rSup { size 8{2} } ={}} {}

= 2 lim x 0 ( 1 2 sin x 2 x 2 ) 2 = 2 1 4 2 lim x 0 ( sin x 2 x 2 ) 2 = 1 2 ( lim x 0 sin x 2 x 2 ) 2 = 1 2 = 2 lim x 0 ( 1 2 sin x 2 x 2 ) 2 = 2 1 4 2 lim x 0 ( sin x 2 x 2 ) 2 = 1 2 ( lim x 0 sin x 2 x 2 ) 2 = 1 2 size 12{ {}=2 {"lim"} cSub { size 8{x rightarrow 0} } $${ {1} over {2} } cdot { {"sin" { {x} over {2} } } over { { {x} over {2} } } }$$ rSup { size 8{2} } =2 cdot { {1} over { { {4} over {2} } } } {"lim"} cSub { size 8{x rightarrow 0} } $${ {"sin" { {x} over {2} } } over { { {x} over {2} } } }$$ rSup { size 8{2} } = { {1} over {2} } $${"lim"} cSub { size 8{x rightarrow 0} } { {"sin" { {x} over {2} } } over { { {x} over {2} } } }$$ rSup { size 8{2} } = { {1} over {2} } } {}
(4)

7. limx01cos3xxsin2x=limx0(1cosx)(1+cosx+cos2x)x2sinxcosx=limx02sin2x2(1+cosx+cos2x)2xcosx2sinx2cosx2=limx01cos3xxsin2x=limx0(1cosx)(1+cosx+cos2x)x2sinxcosx=limx02sin2x2(1+cosx+cos2x)2xcosx2sinx2cosx2= size 12{ {"lim"} cSub { size 8{x rightarrow 0} } { {1 - "cos" rSup { size 8{3} } x} over {x"sin"2x} } = {"lim"} cSub { size 8{x rightarrow 0} } { { $$1 - "cos"x$$ $$1+"cos"x+"cos" rSup { size 8{2} } x$$ } over {x cdot 2"sin"x"cos"x} } = {"lim"} cSub { size 8{x rightarrow 0} } { {2"sin" rSup { size 8{2} } { {x} over {2} } $$1+"cos"x+"cos" rSup { size 8{2} } x$$ } over {2x"cos"x cdot 2"sin" { {x} over {2} } "cos" { {x} over {2} } } } ={}} {}

= lim x 0 sin x 2 ( 1 + cos x + cos 2 x ) 2x cos x cos x 2 == 1 4 lim x 0 tg x 2 ( 1 cos x + 1 + cos x ) x 2 = 1 4 lim x 0 tg x 2 x 2 = 3 4 = lim x 0 sin x 2 ( 1 + cos x + cos 2 x ) 2x cos x cos x 2 == 1 4 lim x 0 tg x 2 ( 1 cos x + 1 + cos x ) x 2 = 1 4 lim x 0 tg x 2 x 2 = 3 4 size 12{ {}= {"lim"} cSub { size 8{x rightarrow 0} } { {"sin" { {x} over {2} } $$1+"cos"x+"cos" rSup { size 8{2} } x$$ } over {2x"cos"x"cos" { {x} over {2} } } } "==" { {1} over {4} } {"lim"} cSub { size 8{x rightarrow 0} } { {"tg" { {x} over {2} } $${ {1} over {"cos"x} } +1+"cos"x$$ } over { { {x} over {2} } } } = { {1} over {4} } {"lim"} cSub { size 8{x rightarrow 0} } { {"tg" { {x} over {2} } } over { { {x} over {2} } } } = { {3} over {4} } } {}
(5)

8. limx01+sinxcosx1sinxcosx=limx02sin2x2+sinx2sin2x2sinx=limx02sin2x2+2sinx2cosx22sin2x22sinx2cosx2=limx01+sinxcosx1sinxcosx=limx02sin2x2+sinx2sin2x2sinx=limx02sin2x2+2sinx2cosx22sin2x22sinx2cosx2= size 12{ {"lim"} cSub { size 8{x rightarrow 0} } { {1+"sin"x - "cos"x} over {1 - "sin"x - "cos"x} } = {"lim"} cSub { size 8{x rightarrow 0} } { {2"sin" rSup { size 8{2} } { {x} over {2} } +"sin"x} over {2"sin" rSup { size 8{2} } { {x} over {2} } - "sin"x} } = {"lim"} cSub { size 8{x rightarrow 0} } { {2"sin" rSup { size 8{2} } { {x} over {2} } +2"sin" { {x} over {2} } "cos" { {x} over {2} } } over {2"sin" rSup { size 8{2} } { {x} over {2} } - 2"sin" { {x} over {2} } "cos" { {x} over {2} } } } ={}} {}

= lim x 0 2 sin x 2 ( sin x 2 + cos x 2 ) 2 sin x 2 ( sin x 2 cos x 2 ) = 0 + 1 0 1 = 1 = lim x 0 2 sin x 2 ( sin x 2 + cos x 2 ) 2 sin x 2 ( sin x 2 cos x 2 ) = 0 + 1 0 1 = 1 size 12{ {}= {"lim"} cSub { size 8{x rightarrow 0} } { {2"sin" { {x} over {2} } $$"sin" { {x} over {2} } +"cos" { {x} over {2} }$$ } over {2"sin" { {x} over {2} } $$"sin" { {x} over {2} } - "cos" { {x} over {2} }$$ } } = { {0+1} over {0 - 1} } = - 1} {}
(6)

9. limx0(1sinx1tgx)=limx0(1sinxcosxsinx)=limx01cosxsin2x=limx0(1sinx1tgx)=limx0(1sinxcosxsinx)=limx01cosxsin2x= size 12{ {"lim"} cSub { size 8{x rightarrow 0} } $${ {1} over {"sin"x} } - { {1} over { ital "tgx"} }$$ = {"lim"} cSub { size 8{x rightarrow 0} } $${ {1} over {"sin"x} } - { {"cos"x} over {"sin"x} }$$ = {"lim"} cSub { size 8{x rightarrow 0} } { {1 - "cos"x} over {"sin" rSup { size 8{2} } x} } ={}} {}

= lim x 0 2 sin 2 x 2 2 sin x 2 cos x 2 = lim x 0 tg x 2 = 0 = lim x 0 2 sin 2 x 2 2 sin x 2 cos x 2 = lim x 0 tg x 2 = 0 size 12{ {}= {"lim"} cSub { size 8{x rightarrow 0} } { {2"sin" rSup { size 8{2} } { {x} over {2} } } over {2"sin" { {x} over {2} } "cos" { {x} over {2} } } } = {"lim"} cSub { size 8{x rightarrow 0} } "tg" { {x} over {2} } =0} {}
(7)

10. limxπ2cosx(1sinx)23=limxπ21sin2x(1sinx)23=limxπ2(1sin2x)3(1sinx)46=limxπ2cosx(1sinx)23=limxπ21sin2x(1sinx)23=limxπ2(1sin2x)3(1sinx)46= size 12{ {"lim"} cSub { size 8{x rightarrow { {π} over {2} } } } { {"cos"x} over { nroot { size 8{3} } { $$1 - "sin"x$$ rSup { size 8{2} } } } } = {"lim"} cSub { size 8{x rightarrow { {π} over {2} } } } { { sqrt {1 - "sin" rSup { size 8{2} } x} } over { nroot { size 8{3} } { $$1 - "sin"x$$ rSup { size 8{2} } } } } = {"lim"} cSub { size 8{x rightarrow { {π} over {2} } } } nroot { size 8{6} } { { { $$1 - "sin" rSup { size 8{2} } x$$ rSup { size 8{3} } } over { $$1 - "sin"x$$ rSup { size 8{4} } } } } ={}} {}

= lim x π 2 ( 1 sin x ) 3 ( 1 + sin x ) 3 ( 1 sin x ) 4 6 = lim x π 2 ( 1 + sin x ) 3 1 sin x 6 = 2 3 0 6 = = lim x π 2 ( 1 sin x ) 3 ( 1 + sin x ) 3 ( 1 sin x ) 4 6 = lim x π 2 ( 1 + sin x ) 3 1 sin x 6 = 2 3 0 6 = size 12{ {}= {"lim"} cSub { size 8{x rightarrow { {π} over {2} } } } nroot { size 8{6} } { { { $$1 - "sin"x$$ rSup { size 8{3} } $$1+"sin"x$$ rSup { size 8{3} } } over { $$1 - "sin"x$$ rSup { size 8{4} } } } } = {"lim"} cSub { size 8{x rightarrow { {π} over {2} } } } nroot { size 8{6} } { { { $$1+"sin"x$$ rSup { size 8{3} } } over {1 - "sin"x} } } = nroot { size 8{6} } { { {2 rSup { size 8{3} } } over {0} } } = infinity } {}
(8)

11. limxπsin3xsin2x=limxπ(32sin3x3xsin2x2x)=32limxπsin3x3xsin2x2x=3211=32limxπsin3xsin2x=limxπ(32sin3x3xsin2x2x)=32limxπsin3x3xsin2x2x=3211=32 size 12{ {"lim"} cSub { size 8{x rightarrow π} } { {"sin"3x} over {"sin"2x} } = {"lim"} cSub { size 8{x rightarrow π} } $${ {3} over {2} } cdot { { { {"sin"3x} over {3x} } } over { { {"sin"2x} over {2x} } } }$$ = { {3} over {2} } cdot {"lim"} cSub { size 8{x rightarrow π} } { { { {"sin"3x} over {3x} } } over { { {"sin"2x} over {2x} } } } = { {3} over {2} } cdot { {1} over {1} } = { {3} over {2} } } {}

12. limx0sin5xsin7x=limx05xsin5x5x7xsin7x7x=57limx0sin5x5xsin7x7x=5711=57limx0sin5xsin7x=limx05xsin5x5x7xsin7x7x=57limx0sin5x5xsin7x7x=5711=57 size 12{ {"lim"} cSub { size 8{x rightarrow 0} } { {"sin"5x} over {"sin"7x} } = {"lim"} cSub { size 8{x rightarrow 0} } { {5x cdot { {"sin"5x} over {5x} } } over {7x cdot { {"sin"7x} over {7x} } } } = { {5} over {7} } {"lim"} cSub { size 8{x rightarrow 0} } { { { {"sin"5x} over {5x} } } over { { {"sin"7x} over {7x} } } } = { {5} over {7} } cdot { {1} over {1} } = { {5} over {7} } } {}.

13. limx0tgxsinxx3=limx0sinxcosxsinxx3=limx0sinxsinxcosxx3cosx=limx0sinx(1cosx)x3cosx=limx0tgxsinxx3=limx0sinxcosxsinxx3=limx0sinxsinxcosxx3cosx=limx0sinx(1cosx)x3cosx= size 12{ {"lim"} cSub { size 8{x rightarrow 0} } { {"tg"x - "sin"x} over {x rSup { size 8{3} } } } = {"lim"} cSub { size 8{x rightarrow 0} } { { { {"sin"x} over {"cos"x} } - "sin"x} over {x rSup { size 8{3} } } } = {"lim"} cSub { size 8{x rightarrow 0} } { {"sin"x - "sin"x"cos"x} over {x rSup { size 8{3} } "cos"x} } = {"lim"} cSub { size 8{x rightarrow 0} } { {"sin"x $$1 - "cos"x$$ } over {x rSup { size 8{3} } "cos"x} } ={}} {}

lim x 0 sin x ( 1 cos x ) x 3 cos x = lim x 0 sin x ( 1 cos x ) x 3 cos x 1 + cos x 1 + cos x = lim x 0 sin x ( 1 cos 2 x ) x 3 cos x ( 1 + cos x ) = lim x 0 sin x ( 1 cos x ) x 3 cos x = lim x 0 sin x ( 1 cos x ) x 3 cos x 1 + cos x 1 + cos x = lim x 0 sin x ( 1 cos 2 x ) x 3 cos x ( 1 + cos x ) = size 12{ {"lim"} cSub { size 8{x rightarrow 0} } { {"sin"x $$1 - "cos"x$$ } over {x rSup { size 8{3} } "cos"x} } = {"lim"} cSub { size 8{x rightarrow 0} } { {"sin"x $$1 - "cos"x$$ } over {x rSup { size 8{3} } "cos"x} } cdot { {1+"cos"x} over {1+"cos"x} } = {"lim"} cSub { size 8{x rightarrow 0} } { {"sin"x $$1 - "cos" rSup { size 8{2} } x$$ } over {x rSup { size 8{3} } "cos"x cdot $$1+"cos"x$$ } } ={}} {}
(9)

=limx0sin3xx3cosx(1+cosx)=limx0sin3xx3limx01cosx(1+cosx)=1311(1+1)=12=limx0sin3xx3cosx(1+cosx)=limx0sin3xx3limx01cosx(1+cosx)=1311(1+1)=12 size 12{ {}= {"lim"} cSub { size 8{x rightarrow 0} } { {"sin" rSup { size 8{3} } x} over {x rSup { size 8{3} } "cos"x $$1+"cos"x$$ } } = {"lim"} cSub { size 8{x rightarrow 0} } { {"sin" rSup { size 8{3} } x} over {x rSup { size 8{3} } } } cdot {"lim"} cSub { size 8{x rightarrow 0} } { {1} over {"cos"x $$1+"cos"x$$ } } =1 rSup { size 8{3} } cdot { {1} over {1 cdot $$1+1$$ } } = { {1} over {2} } } {}.

14. limxπ2(π2x)tgx=limxπ2(π2x)sinxcosx=limxπ2(π2x)sinxsin(π2x)=limxπ2sinxsin(π2x)π2x=11=1limxπ2(π2x)tgx=limxπ2(π2x)sinxcosx=limxπ2(π2x)sinxsin(π2x)=limxπ2sinxsin(π2x)π2x=11=1 size 12{ {"lim"} cSub { size 8{x rightarrow { {π} over {2} } } } $${ {π} over {2} } - x$$ "tg"x= {"lim"} cSub { size 8{x rightarrow { {π} over {2} } } } { { $${ {π} over {2} } - x$$ "sin"x} over {"cos"x} } = {"lim"} cSub { size 8{x rightarrow { {π} over {2} } } } { { $${ {π} over {2} } - x$$ "sin"x} over {"sin" $${ {π} over {2} } - x$$ } } = {"lim"} cSub { size 8{x rightarrow { {π} over {2} } } } { {"sin"x} over { { {"sin" $${ {π} over {2} } - x$$ } over { { {π} over {2} } - x} } } } = { {1} over {1} } =1} {}

15. limxπsinx1x2π2=limxπcosx0π2=12π=π2limxπsinx1x2π2=limxπcosx0π2=12π=π2 size 12{ {"lim"} cSub { size 8{x rightarrow π} } { {"sin"x} over {1 - { {x rSup { size 8{2} } } over {π rSup { size 8{2} } } } } } = {"lim"} cSub { size 8{x rightarrow π} } { {"cos"x} over {0 - { {π} over {2} } } } = { { - 1} over { - { {2} over {π} } } } = { {π} over {2} } } {}

16. limz1(1z)tgπz2=limz1(1z)sinπz2cosπz2=limz11zcosπz2limz1sinπz2=limz1(1z)tgπz2=limz1(1z)sinπz2cosπz2=limz11zcosπz2limz1sinπz2= size 12{ {"lim"} cSub { size 8{z rightarrow 1} } $$1 - z$$ "tg" { {πz} over {2} } = {"lim"} cSub { size 8{z rightarrow 1} } { { $$1 - z$$ "sin" { {πz} over {2} } } over {"cos" { {πz} over {2} } } } = {"lim"} cSub { size 8{z rightarrow 1} } { {1 - z} over {"cos" { {πz} over {2} } } } {"lim"} cSub { size 8{z rightarrow 1} } "sin" { {πz} over {2} } ={}} {}

= lim z 1 1 z sin ( π 2 πz 2 ) 1 = lim z 1 1 z sin π 2 ( 1 z ) = lim z 1 2 π π 2 ( 1 z ) sin π 2 ( 1 z ) = = lim z 1 1 z sin ( π 2 πz 2 ) 1 = lim z 1 1 z sin π 2 ( 1 z ) = lim z 1 2 π π 2 ( 1 z ) sin π 2 ( 1 z ) = size 12{ {}= {"lim"} cSub { size 8{z rightarrow 1} } { {1 - z} over {"sin" $${ {π} over {2} } - { {πz} over {2} }$$ } } 1= {"lim"} cSub { size 8{z rightarrow 1} } { {1 - z} over {"sin" { {π} over {2} } $$1 - z$$ } } = {"lim"} cSub { size 8{z rightarrow 1} } { {2} over {π} } { { { {π} over {2} } $$1 - z$$ } over {"sin" { {π} over {2} } $$1 - z$$ } } ={}} {}
(10)
= 2 π 1 lim z 1 sin π 2 ( 1 z ) π 2 ( 1 z ) = 2 π = 2 π 1 lim z 1 sin π 2 ( 1 z ) π 2 ( 1 z ) = 2 π size 12{ {}= { {2} over {π} } { {1} over { {"lim"} cSub { size 8{z rightarrow 1} } { {"sin" { {π} over {2} } $$1 - z$$ } over { { {π} over {2} } $$1 - z$$ } } } } = { {2} over {π} } } {}
(11)

17. limxπ6sin(xπ6)32cosx=limxπ6cos(xπ6)sinx=112=2limxπ6sin(xπ6)32cosx=limxπ6cos(xπ6)sinx=112=2 size 12{ {"lim"} cSub { size 8{x rightarrow { {π} over {6} } } } { {"sin" $$x - { {π} over {6} }$$ } over { { { sqrt {3} } over {2} } - "cos"x} } = {"lim"} cSub { size 8{x rightarrow { {π} over {6} } } } { {"cos" $$x - { {π} over {6} }$$ } over {"sin"x} } = { {1} over { { {1} over {2} } } } =2} {}

18. limxπ1sinx2cosx2(cosx4sinx4)=limxπ(1sinx2)(1+sinx2)cosx2(cosx4sinx4)(1+sinx2)=limxπ1sinx2cosx2(cosx4sinx4)=limxπ(1sinx2)(1+sinx2)cosx2(cosx4sinx4)(1+sinx2)= size 12{ {"lim"} cSub { size 8{x rightarrow π} } { {1 - "sin" { {x} over {2} } } over {"cos" { {x} over {2} } $$"cos" { {x} over {4} } - "sin" { {x} over {4} }$$ } } = {"lim"} cSub { size 8{x rightarrow π} } { { $$1 - "sin" { {x} over {2} }$$ $$1+"sin" { {x} over {2} }$$ } over {"cos" { {x} over {2} } $$"cos" { {x} over {4} } - "sin" { {x} over {4} }$$ $$1+"sin" { {x} over {2} }$$ } } ={}} {}

= lim x π 1 sin 2 x 2 2 cos x 2 ( cos x 4 sin x 4 ) = lim x π cos 2 x 2 2 cos x 2 ( cos x 4 sin x 4 ) = = lim x π 1 sin 2 x 2 2 cos x 2 ( cos x 4 sin x 4 ) = lim x π cos 2 x 2 2 cos x 2 ( cos x 4 sin x 4 ) = size 12{ {}= {"lim"} cSub { size 8{x rightarrow π} } { {1 - "sin" rSup { size 8{2} } { {x} over {2} } } over {2"cos" { {x} over {2} } $$"cos" { {x} over {4} } - "sin" { {x} over {4} }$$ } } = {"lim"} cSub { size 8{x rightarrow π} } { {"cos" rSup { size 8{2} } { {x} over {2} } } over {2"cos" { {x} over {2} } $$"cos" { {x} over {4} } - "sin" { {x} over {4} }$$ } } ={}} {} = lim x π cos x 2 ( cos x 4 + sin x 4 ) 2 ( cos x 4 sin x 4 ) ( cos x 4 + sin x 4 ) = lim x π cos x 2 ( cos x 4 + sin x 4 ) 2 ( cos 2 x 4 sin 2 x 4 ) = = lim x π cos x 2 ( cos x 4 + sin x 4 ) 2 ( cos x 4 sin x 4 ) ( cos x 4 + sin x 4 ) = lim x π cos x 2 ( cos x 4 + sin x 4 ) 2 ( cos 2 x 4 sin 2 x 4 ) = size 12{ {}= {"lim"} cSub { size 8{x rightarrow π} } { {"cos" { {x} over {2} } $$"cos" { {x} over {4} } +"sin" { {x} over {4} }$$ } over {2 $$"cos" { {x} over {4} } - "sin" { {x} over {4} }$$ $$"cos" { {x} over {4} } +"sin" { {x} over {4} }$$ } } = {"lim"} cSub { size 8{x rightarrow π} } { {"cos" { {x} over {2} } $$"cos" { {x} over {4} } +"sin" { {x} over {4} }$$ } over {2 $$"cos" rSup { size 8{2} } { {x} over {4} } - "sin" rSup { size 8{2} } { {x} over {4} }$$ } } ={}} {}

= lim x π cos x 2 ( cos x 4 + sin x 4 ) 2 cos x 2 = cos π 4 + sin π 4 2 = 2 2 + 2 2 2 = 2 2 = lim x π cos x 2 ( cos x 4 + sin x 4 ) 2 cos x 2 = cos π 4 + sin π 4 2 = 2 2 + 2 2 2 = 2 2 size 12{ {}= {"lim"} cSub { size 8{x rightarrow π} } { {"cos" { {x} over {2} } $$"cos" { {x} over {4} } +"sin" { {x} over {4} }$$ } over {2"cos" { {x} over {2} } } } = { {"cos" { {π} over {4} } +"sin" { {π} over {4} } } over {2} } = { { { { sqrt {2} } over {2} } + { { sqrt {2} } over {2} } } over {2} } = { { sqrt {2} } over {2} } } {}
(12)

19. limx0cosaxcosbxx2=limx02sinax+bx2sinaxbx2x2=limx0cosaxcosbxx2=limx02sinax+bx2sinaxbx2x2= size 12{ {"lim"} cSub { size 8{x rightarrow 0} } { {"cos" ital "ax" - "cos" ital "bx"} over {x rSup { size 8{2} } } } = {"lim"} cSub { size 8{x rightarrow 0} } { { - 2"sin" { { ital "ax"+ ital "bx"} over {2} } "sin" { { ital "ax" - ital "bx"} over {2} } } over {x rSup { size 8{2} } } } ={}} {}=2limx0(sinx(a+b)2xsinx(ab)2x)==2limx0(sinx(a+b)2xsinx(ab)2x)= size 12{ {}= - 2 {"lim"} cSub { size 8{x rightarrow 0} } $${ {"sin" { {x \( a+b$$ } over {2} } } over {x} } cdot { {"sin" { {x $$a - b$$ } over {2} } } over {x} } \) ={}} {}

= 2 lim x 0 ( a + b 2 sin x ( a + b ) 2 ( a + b ) x 2 a b 2 sin x ( a b 2 ) ( a b ) 2 x ) = 2 a + b 2 1 a b 2 1 = = 2 lim x 0 ( a + b 2 sin x ( a + b ) 2 ( a + b ) x 2 a b 2 sin x ( a b 2 ) ( a b ) 2 x ) = 2 a + b 2 1 a b 2 1 = size 12{ {}= - 2 {"lim"} cSub { size 8{x rightarrow 0} } $${ {a+b} over {2} } cdot { {"sin" { {x \( a+b$$ } over {2} } } over { { { $$a+b$$ x} over {2} } } } cdot { {a - b} over {2} } cdot { {"sin"x $${ {a - b} over {2} }$$ } over { { { $$a - b$$ } over {2} } x} } \) = - 2 cdot { {a+b} over {2} } cdot 1 cdot { {a - b} over {2} } cdot 1={}} {}
(13)
= 2 a 2 b 2 4 = a 2 b 2 2 = b 2 a 2 2 = 2 a 2 b 2 4 = a 2 b 2 2 = b 2 a 2 2 size 12{ {}= - 2 cdot { {a rSup { size 8{2} } - b rSup { size 8{2} } } over {4} } = - { {a rSup { size 8{2} } - b rSup { size 8{2} } } over {2} } = { {b rSup { size 8{2} } - a rSup { size 8{2} } } over {2} } } {}
(14)

20. limx021+cosxsin2x=limx022cos2x2sin2x=limx022cosx2sin2x=limx021+cosxsin2x=limx022cos2x2sin2x=limx022cosx2sin2x= size 12{ {"lim"} cSub { size 8{x rightarrow 0} } { { sqrt {2} - sqrt {1+"cos"x} } over {"sin" rSup { size 8{2} } x} } = {"lim"} cSub { size 8{x rightarrow 0} } { { sqrt {2} - sqrt {2"cos" rSup { size 8{2} } { {x} over {2} } } } over {"sin" rSup { size 8{2} } x} } = {"lim"} cSub { size 8{x rightarrow 0} } { { sqrt {2} - sqrt {2} "cos" { {x} over {2} } } over {"sin" rSup { size 8{2} } x} } ={}} {}

= lim x 0 2 ( 1 cos x 2 ) sin 2 x = lim x 0 2 2 sin 2 x 4 2 2 sin 2 x 2 cos 2 x 2 = lim x 0 2 ( 1 cos x 2 ) sin 2 x = lim x 0 2 2 sin 2 x 4 2 2 sin 2 x 2 cos 2 x 2 size 12{ {}= {"lim"} cSub { size 8{x rightarrow 0} } { { sqrt {2} $$1 - "cos" { {x} over {2} }$$ } over {"sin" rSup { size 8{2} } x} } = {"lim"} cSub { size 8{x rightarrow 0} } { { sqrt {2} cdot 2"sin" rSup { size 8{2} } { {x} over {4} } } over {2 rSup { size 8{2} } "sin" rSup { size 8{2} } { {x} over {2} } "cos" rSup { size 8{2} } { {x} over {2} } } } } {}
(15)
= 2 2 lim x 0 sin 2 x 4 2 2 sin 2 x 4 cos 2 x 4 cos 2 x 2 = 2 2 1 4 1 1 = 2 8 = 2 2 lim x 0 sin 2 x 4 2 2 sin 2 x 4 cos 2 x 4 cos 2 x 2 = 2 2 1 4 1 1 = 2 8 size 12{ {}= { { sqrt {2} } over {2} } {"lim"} cSub { size 8{x rightarrow 0} } { {"sin" rSup { size 8{2} } { {x} over {4} } } over {2 rSup { size 8{2} } cdot "sin" rSup { size 8{2} } { {x} over {4} } "cos" rSup { size 8{2} } { {x} over {4} } "cos" rSup { size 8{2} } { {x} over {2} } } } = { { sqrt {2} } over {2} } cdot { {1} over {4 cdot 1 cdot 1} } = { { sqrt {2} } over {8} } } {}
(16)

21. limx01+sinx1sinxtgx1+sinx+1sinx1+sinx+1sinx=limx01+sinx1sinxtgx1+sinx+1sinx1+sinx+1sinx= size 12{ {"lim"} cSub { size 8{x rightarrow 0} } { { sqrt {1+"sin"x} - sqrt {1 - "sin"x} } over {"tg"x} } cdot { { sqrt {1+"sin"x} + sqrt {1 - "sin"x} } over { sqrt {1+"sin"x} + sqrt {1 - "sin"x} } } ={}} {}

= lim x 0 ( 1 + sin x ) 2 ( 1 sin x ) 2 sin x cos x ( 1 + sin x + 1 sin x ) = = lim x 0 ( 1 + sin x ) 2 ( 1 sin x ) 2 sin x cos x ( 1 + sin x + 1 sin x ) = size 12{ {}= {"lim"} cSub { size 8{x rightarrow 0} } { { $$sqrt {1+"sin"x$$ } rSup { size 8{2} } - $$sqrt {1 - "sin"x$$ } rSup { size 8{2} } } over { { {"sin"x} over {"cos"x} } $$sqrt {1+"sin"x} + sqrt {1 - "sin"x$$ } } } ={}} {}
(17)
= lim x 0 1 + sin x 1 + sin x sin x cos x ( 1 + sin x + 1 sin x ) = lim x 0 2 sin x sin x cos x ( 1 + sin x + 1 sin x ) = = lim x 0 1 + sin x 1 + sin x sin x cos x ( 1 + sin x + 1 sin x ) = lim x 0 2 sin x sin x cos x ( 1 + sin x + 1 sin x ) = size 12{ {}= {"lim"} cSub { size 8{x rightarrow 0} } { {1+"sin"x - 1+"sin"x} over { { {"sin"x} over {"cos"x} } $$sqrt {1+"sin"x} + sqrt {1 - "sin"x$$ } } } = {"lim"} cSub { size 8{x rightarrow 0} } { {2"sin"x} over { { {"sin"x} over {"cos"x} } $$sqrt {1+"sin"x} + sqrt {1 - "sin"x$$ } } } ={}} {}
(18)
= lim x 0 2 cos x 1 + sin x + 1 sin x = 2 1 + 1 = 2 2 = 1 = lim x 0 2 cos x 1 + sin x + 1 sin x = 2 1 + 1 = 2 2 = 1 size 12{ {}= {"lim"} cSub { size 8{x rightarrow 0} } { {2"cos"x} over { sqrt {1+"sin"x} + sqrt {1 - "sin"x} } } = { {2} over {1+1} } = { {2} over {2} } =1} {}
(19)

22. limx01+xsinxcos2xtan2x21+xsinx+cos2x1+xsinx+cos2x=limx0(1+xsinx)2(cos2x)2tan2x2(1+xsinx+cos2x)=limx01+xsinxcos2xtan2x21+xsinx+cos2x1+xsinx+cos2x=limx0(1+xsinx)2(cos2x)2tan2x2(1+xsinx+cos2x)= size 12{ {"lim"} cSub { size 8{x rightarrow 0} } { { sqrt {1+x"sin"x} - sqrt {"cos"2x} } over {"tan" rSup { size 8{2} } { {x} over {2} } } } { { sqrt {1+x"sin"x} + sqrt {"cos"2x} } over { sqrt {1+x"sin"x} + sqrt {"cos"2x} } } = {"lim"} cSub { size 8{x rightarrow 0} } { { $$sqrt {1+x"sin"x}$$ rSup { size 8{2} } - $$sqrt {"cos"2x}$$ rSup { size 8{2} } } over {"tan" rSup { size 8{2} } { {x} over {2} } $$sqrt {1+x"sin"x} + sqrt {"cos"2x}$$ } } ={}} {}

= lim x 0 sin 2 x + cos 2 x + x sin x cos 2 x + sin 2 x sin 2 x 2 cos 2 x 2 ( 1 + x sin x + cos 2x ) = = lim x 0 sin 2 x + cos 2 x + x sin x cos 2 x + sin 2 x sin 2 x 2 cos 2 x 2 ( 1 + x sin x + cos 2x ) = size 12{ {}= {"lim"} cSub { size 8{x rightarrow 0} } { {"sin" rSup { size 8{2} } x+"cos" rSup { size 8{2} } x+x"sin"x - "cos" rSup { size 8{2} } x+"sin" rSup { size 8{2} } x} over { { {"sin" rSup { size 8{2} } { {x} over {2} } } over {"cos" rSup { size 8{2} } { {x} over {2} } } } $$sqrt {1+x"sin"x} + sqrt {"cos"2x}$$ } } ={}} {}
(20)
= lim x 0 sin x ( 2 sin x + x ) sin 2 x 2 cos 2 x 2 ( 1 + x sin x + cos 2x ) = lim x 0 2 sin x 2 cos x 2 ( 2 sin x + x ) sin 2 x 2 cos 2 x 2 ( 1 + x sin x + cos 2x ) = = lim x 0 sin x ( 2 sin x + x ) sin 2 x 2 cos 2 x 2 ( 1 + x sin x + cos 2x ) = lim x 0 2 sin x 2 cos x 2 ( 2 sin x + x ) sin 2 x 2 cos 2 x 2 ( 1 + x sin x + cos 2x ) = size 12{ {}= {"lim"} cSub { size 8{x rightarrow 0} } { {"sin"x $$2"sin"x+x$$ } over { { {"sin" rSup { size 8{2} } { {x} over {2} } } over {"cos" rSup { size 8{2} } { {x} over {2} } } } $$sqrt {1+x"sin"x} + sqrt {"cos"2x}$$ } } = {"lim"} cSub { size 8{x rightarrow 0} } { {2"sin" { {x} over {2} } "cos" { {x} over {2} } $$2"sin"x+x$$ } over { { {"sin" rSup { size 8{2} } { {x} over {2} } } over {"cos" rSup { size 8{2} } { {x} over {2} } } } $$sqrt {1+x"sin"x} + sqrt {"cos"2x}$$ } } ={}} {}
(21)
= lim x 0 2 ( 2 sin x + x ) 1 + x sin x + cos 2x cos x 2 sin x 2 = lim x 0 2 cos x 2 1 + x sin x + cos 2x ( 4 sin x 2 cos x 2 sin x 2 + x sin x 2 ) = = lim x 0 2 ( 2 sin x + x ) 1 + x sin x + cos 2x cos x 2 sin x 2 = lim x 0 2 cos x 2 1 + x sin x + cos 2x ( 4 sin x 2 cos x 2 sin x 2 + x sin x 2 ) = size 12{ {}= {"lim"} cSub { size 8{x rightarrow 0} } { {2 $$2"sin"x+x$$ } over { sqrt {1+x"sin"x} + sqrt {"cos"2x} } } { {"cos" { {x} over {2} } } over {"sin" { {x} over {2} } } } = {"lim"} cSub { size 8{x rightarrow 0} } { {2"cos" { {x} over {2} } } over { sqrt {1+x"sin"x} + sqrt {"cos"2x} } } $${ {4"sin" { {x} over {2} } "cos" { {x} over {2} } } over {"sin" { {x} over {2} } } } + { {x} over {"sin" { {x} over {2} } } }$$ ={}} {}
(22)
= lim x 0 2 cos x 2 1 + x sin x + cos 2x ( 4 cos x 2 + 1 sin x 2 2 x 2 ) = 2 2 ( 4 + 2 ) = 6 = lim x 0 2 cos x 2 1 + x sin x + cos 2x ( 4 cos x 2 + 1 sin x 2 2 x 2 ) = 2 2 ( 4 + 2 ) = 6 size 12{ {}= {"lim"} cSub { size 8{x rightarrow 0} } { {2"cos" { {x} over {2} } } over { sqrt {1+x"sin"x} + sqrt {"cos"2x} } } $$4"cos" { {x} over {2} } + { {1} over { { {"sin" { {x} over {2} } } over {2 cdot { {x} over {2} } } } } }$$ = { {2} over {2} } $$4+2$$ =6} {}
(23)

23. limxπ4=cosxsinxcos2x=limxπ4cosxsinxcos2xsin2x=limxπ4cosxsinx(cosxsinx)(cosxsinx)=limxπ4=cosxsinxcos2x=limxπ4cosxsinxcos2xsin2x=limxπ4cosxsinx(cosxsinx)(cosxsinx)= size 12{ {"lim"} cSub { size 8{x rightarrow { {π} over {4} } } } = { {"cos"x - "sin"x} over {"cos"2x} } = {"lim"} cSub { size 8{x rightarrow { {π} over {4} } } } { {"cos"x - "sin"x} over {"cos" rSup { size 8{2} } x - "sin" rSup { size 8{2} } x} } = {"lim"} cSub { size 8{x rightarrow { {π} over {4} } } } { {"cos"x - "sin"x} over { $$"cos"x - "sin"x$$ $$"cos"x - "sin"x$$ } } ={}} {}

= lim x π 4 1 cos x sin x = 1 cos π 4 + sin π 4 = 1 2 2 + 2 2 = 1 2 2 2 = 2 2 = lim x π 4 1 cos x sin x = 1 cos π 4 + sin π 4 = 1 2 2 + 2 2 = 1 2 2 2 = 2 2 size 12{ {}= {"lim"} cSub { size 8{x rightarrow { {π} over {4} } } } { {1} over {"cos"x - "sin"x} } = { {1} over {"cos" { {π} over {4} } +"sin" { {π} over {4} } } } = { {1} over { { { sqrt {2} } over {2} } + { { sqrt {2} } over {2} } } } = { {1} over { sqrt {2} } } cdot { { sqrt {2} } over { sqrt {2} } } = { { sqrt {2} } over {2} } } {}
(24)

24. limx01cosxcos2xx21+cosxcos2x1+cosxcos2x=sin2x+cos2xcos2x(cos2xsin2x)x2(1+cosxcos2x)=limx01cosxcos2xx21+cosxcos2x1+cosxcos2x=sin2x+cos2xcos2x(cos2xsin2x)x2(1+cosxcos2x)= size 12{ {"lim"} cSub { size 8{x rightarrow 0} } { {1 - "cos"x sqrt {"cos"2x} } over {x rSup { size 8{2} } } } { {1+"cos"x sqrt {"cos"2x} } over {1+"cos"x sqrt {"cos"2x} } } = { {"sin" rSup { size 8{2} } x+"cos" rSup { size 8{2} } x - "cos" rSup { size 8{2} } x $$"cos" rSup { size 8{2} } x - "sin" rSup { size 8{2} } x$$ } over {x rSup { size 8{2} } $$1+"cos"x sqrt {"cos"2x$$ } } } ={}} {}

= lim x 0 sin 2 x + cos 2 x cos 4 x + sin 2 x cos 2 x x ( 1 + cos x cos 2x ) = lim x 0 sin 2 x ( 1 + cos 2 x ) + cos 2 x ( 1 cos x ) x 2 ( 1 + cos x cos 2x ) = = lim x 0 sin 2 x + cos 2 x cos 4 x + sin 2 x cos 2 x x ( 1 + cos x cos 2x ) = lim x 0 sin 2 x ( 1 + cos 2 x ) + cos 2 x ( 1 cos x ) x 2 ( 1 + cos x cos 2x ) = size 12{ {}= {"lim"} cSub { size 8{x rightarrow 0} } { {"sin" rSup { size 8{2} } x+"cos" rSup { size 8{2} } x - "cos" rSup { size 8{4} } x+"sin" rSup { size 8{2} } x"cos" rSup { size 8{2} } x} over {x $$1+"cos"x sqrt {"cos"2x$$ } } } = {"lim"} cSub { size 8{x rightarrow 0} } { {"sin" rSup { size 8{2} } x $$1+"cos" rSup { size 8{2} } x$$ +"cos" rSup { size 8{2} } x $$1 - "cos"x$$ } over {x rSup { size 8{2} } $$1+"cos"x sqrt {"cos"2x$$ } } } ={}} {}
(25)

= lim x 0 sin 2 x ( 1 + cos 2 x ) + cos 2 x sin 2 x x 2 ( 1 + cos x cos 2x ) = lim x 0 sin 2 x ( 1 + 2 cos 2 x ) x 2 ( 1 + cos x cos 2x ) = 1 + 2 1 + 1 = 3 2 = lim x 0 sin 2 x ( 1 + cos 2 x ) + cos 2 x sin 2 x x 2 ( 1 + cos x cos 2x ) = lim x 0 sin 2 x ( 1 + 2 cos 2 x ) x 2 ( 1 + cos x cos 2x ) = 1 + 2 1 + 1 = 3 2 size 12{ {}= {"lim"} cSub { size 8{x rightarrow 0} } { {"sin" rSup { size 8{2} } x $$1+"cos" rSup { size 8{2} } x$$ +"cos" rSup { size 8{2} } x"sin" rSup { size 8{2} } x} over {x rSup { size 8{2} } $$1+"cos"x sqrt {"cos"2x$$ } } } = {"lim"} cSub { size 8{x rightarrow 0} } { {"sin" rSup { size 8{2} } x $$1+2"cos" rSup { size 8{2} } x$$ } over {x rSup { size 8{2} } $$1+"cos"x sqrt {"cos"2x$$ } } } = { {1+2} over {1+1} } = { {3} over {2} } } {} {}

25. limxBsin2xsin2Bx2B2=limxB(sinxsinB)(sinx+sinB)(xB)(x+B)=limxBsinx+B(x+B)=sin2B2BlimxBsin2xsin2Bx2B2=limxB(sinxsinB)(sinx+sinB)(xB)(x+B)=limxBsinx+B(x+B)=sin2B2B size 12{ {"lim"} cSub { size 8{x rightarrow B} } { {"sin" rSup { size 8{2} } x - "sin" rSup { size 8{2} } B} over {x rSup { size 8{2} } - B rSup { size 8{2} } } } = {"lim"} cSub { size 8{x rightarrow B} } { { $$"sin"x - "sin"B$$ $$"sin"x+"sin"B$$ } over { $$x - B$$ $$x+B$$ } } = {"lim"} cSub { size 8{x rightarrow B} } { {"sin"x+B} over { $$x+B$$ } } = { {"sin"2B} over {2B} } } {}

26. limx02arcsinx3x=limx02t3sint=23limx01sintt=23limx02arcsinx3x=limx02t3sint=23limx01sintt=23 size 12{ {"lim"} cSub { size 8{x rightarrow 0} } { {2"arcsin"x} over {3x} } = {"lim"} cSub { size 8{x rightarrow 0} } { {2t} over {3"sin"t} } = { {2} over {3} } cdot {"lim"} cSub { size 8{x rightarrow 0} } { {1} over { { {"sin"t} over {t} } } } = { {2} over {3} } } {}

смена: sint=x,x0t0sint=x,x0t0 size 12{"sin"t=x,~x rightarrow 0 drarrow t rightarrow 0} {}

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