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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. limx→0sin3xx=limx→0cos2xsin3xx=limx→0cosxsin3x3x3xxx=limx→0sin3xx=limx→0cos2xsin3xx=limx→0cosxsin3x3x3xxx= 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. limx→0sinαxsinβx=limx→0αsinαxαxβsinβxβx=αβ⋅sin0sin0=0limx→0sinαxsinβx=limx→0αsinαxαxβsinβxβx=αβ⋅sin0sin0=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. limx→0tgkxx=limx→0sinkxcoskxx=limx→0sinkxxcoskx⋅kk=limx→0(ksinkxkx1coskx)=klimx→0tgkxx=limx→0sinkxcoskxx=limx→0sinkxxcoskx⋅kk=limx→0(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. limx→0tan2xsin5x=limx→0sin2xcos2xsin5x=limx→02⋅5xsin2x2⋅5xcos2xsin5x=limx→0tan2xsin5x=limx→0sin2xcos2xsin5x=limx→02⋅5xsin2x2⋅5xcos2xsin5x= 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. limx→0sin(xn)(sinx)m=limx→0sin(xn)sinmx=limx→0sin(xn)xnsinmxxn=limx→0sin(xn)(sinx)m=limx→0sin(xn)sinmx=limx→0sin(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. limx→01−cosxx2=limx→02sin2x2x2=2limx→0(sinx2x)2=2limx→0(sinx22⋅x2)2=limx→01−cosxx2=limx→02sin2x2x2=2limx→0(sinx2x)2=2limx→0(sinx22⋅x2)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. limx→01−cos3xxsin2x=limx→0(1−cosx)(1+cosx+cos2x)x⋅2sinxcosx=limx→02sin2x2(1+cosx+cos2x)2xcosx⋅2sinx2cosx2=limx→01−cos3xxsin2x=limx→0(1−cosx)(1+cosx+cos2x)x⋅2sinxcosx=limx→02sin2x2(1+cosx+cos2x)2xcosx⋅2sinx2cosx2= 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. limx→01+sinx−cosx1−sinx−cosx=limx→02sin2x2+sinx2sin2x2−sinx=limx→02sin2x2+2sinx2cosx22sin2x2−2sinx2cosx2=limx→01+sinx−cosx1−sinx−cosx=limx→02sin2x2+sinx2sin2x2−sinx=limx→02sin2x2+2sinx2cosx22sin2x2−2sinx2cosx2= 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. limx→0(1sinx−1tgx)=limx→0(1sinx−cosxsinx)=limx→01−cosxsin2x=limx→0(1sinx−1tgx)=limx→0(1sinx−cosxsinx)=limx→01−cosxsin2x= 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. limx→π2cosx(1−sinx)23=limx→π21−sin2x(1−sinx)23=limx→π2(1−sin2x)3(1−sinx)46=limx→π2cosx(1−sinx)23=limx→π21−sin2x(1−sinx)23=limx→π2(1−sin2x)3(1−sinx)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. limx→πsin3xsin2x=limx→π(32⋅sin3x3xsin2x2x)=32⋅limx→πsin3x3xsin2x2x=32⋅11=32limx→πsin3xsin2x=limx→π(32⋅sin3x3xsin2x2x)=32⋅limx→πsin3x3xsin2x2x=32⋅11=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. limx→0sin5xsin7x=limx→05x⋅sin5x5x7x⋅sin7x7x=57limx→0sin5x5xsin7x7x=57⋅11=57limx→0sin5xsin7x=limx→05x⋅sin5x5x7x⋅sin7x7x=57limx→0sin5x5xsin7x7x=57⋅11=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. limx→0tgx−sinxx3=limx→0sinxcosx−sinxx3=limx→0sinx−sinxcosxx3cosx=limx→0sinx(1−cosx)x3cosx=limx→0tgx−sinxx3=limx→0sinxcosx−sinxx3=limx→0sinx−sinxcosxx3cosx=limx→0sinx(1−cosx)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)

=limx→0sin3xx3cosx(1+cosx)=limx→0sin3xx3⋅limx→01cosx(1+cosx)=13⋅11⋅(1+1)=12=limx→0sin3xx3cosx(1+cosx)=limx→0sin3xx3⋅limx→01cosx(1+cosx)=13⋅11⋅(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. limx→π2(π2−x)tgx=limx→π2(π2−x)sinxcosx=limx→π2(π2−x)sinxsin(π2−x)=limx→π2sinxsin(π2−x)π2−x=11=1limx→π2(π2−x)tgx=limx→π2(π2−x)sinxcosx=limx→π2(π2−x)sinxsin(π2−x)=limx→π2sinxsin(π2−x)π2−x=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. limx→πsinx1−x2π2=limx→πcosx0−π2=−1−2π=π2limx→πsinx1−x2π2=limx→πcosx0−π2=−1−2π=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. limz→1(1−z)tgπz2=limz→1(1−z)sinπz2cosπz2=limz→11−zcosπz2limz→1sinπz2=limz→1(1−z)tgπz2=limz→1(1−z)sinπz2cosπz2=limz→11−zcosπz2limz→1sinπz2= 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. limx→π6sin(x−π6)32−cosx=limx→π6cos(x−π6)sinx=112=2limx→π6sin(x−π6)32−cosx=limx→π6cos(x−π6)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. limx→π1−sinx2cosx2(cosx4−sinx4)=limx→π(1−sinx2)(1+sinx2)cosx2(cosx4−sinx4)(1+sinx2)=limx→π1−sinx2cosx2(cosx4−sinx4)=limx→π(1−sinx2)(1+sinx2)cosx2(cosx4−sinx4)(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. limx→0cosax−cosbxx2=limx→0−2sinax+bx2sinax−bx2x2=limx→0cosax−cosbxx2=limx→0−2sinax+bx2sinax−bx2x2= 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} } } } ={}} {}=−2limx→0(sinx(a+b)2x⋅sinx(a−b)2x)==−2limx→0(sinx(a+b)2x⋅sinx(a−b)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. limx→02−1+cosxsin2x=limx→02−2cos2x2sin2x=limx→02−2cosx2sin2x=limx→02−1+cosxsin2x=limx→02−2cos2x2sin2x=limx→02−2cosx2sin2x= 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. limx→01+sinx−1−sinxtgx⋅1+sinx+1−sinx1+sinx+1−sinx=limx→01+sinx−1−sinxtgx⋅1+sinx+1−sinx1+sinx+1−sinx= 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. limx→01+xsinx−cos2xtan2x21+xsinx+cos2x1+xsinx+cos2x=limx→0(1+xsinx)2−(cos2x)2tan2x2(1+xsinx+cos2x)=limx→01+xsinx−cos2xtan2x21+xsinx+cos2x1+xsinx+cos2x=limx→0(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. limx→π4=cosx−sinxcos2x=limx→π4cosx−sinxcos2x−sin2x=limx→π4cosx−sinx(cosx−sinx)(cosx−sinx)=limx→π4=cosx−sinxcos2x=limx→π4cosx−sinxcos2x−sin2x=limx→π4cosx−sinx(cosx−sinx)(cosx−sinx)= 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. limx→01−cosxcos2xx21+cosxcos2x1+cosxcos2x=sin2x+cos2x−cos2x(cos2x−sin2x)x2(1+cosxcos2x)=limx→01−cosxcos2xx21+cosxcos2x1+cosxcos2x=sin2x+cos2x−cos2x(cos2x−sin2x)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. limx→Bsin2x−sin2Bx2−B2=limx→B(sinx−sinB)(sinx+sinB)(x−B)(x+B)=limx→Bsinx+B(x+B)=sin2B2Blimx→Bsin2x−sin2Bx2−B2=limx→B(sinx−sinB)(sinx+sinB)(x−B)(x+B)=limx→Bsinx+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. limx→02arcsinx3x=limx→02t3sint=23⋅limx→01sintt=23limx→02arcsinx3x=limx→02t3sint=23⋅limx→01sintt=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,x→0⇒t→0sint=x,x→0⇒t→0 size 12{"sin"t=x,~x rightarrow 0 drarrow t rightarrow 0} {}

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