Лимеси со Лопиталово правило

1. limx→1ex−ex−1=ЛПlimx→1(ex−e)'(x−1)'=limx→1ex=e1=elimx→1ex−ex−1=ЛПlimx→1(ex−e)'(x−1)'=limx→1ex=e1=e size 12{ {"lim"} cSub { size 8{x rightarrow 1} } { {e rSup { size 8{x} } - e} over {x - 1} } { size 24{ {}={}} } cSup { size 8{"ЛП"} } {"lim"} cSub { size 8{x rightarrow 1} } { { \( e rSup { size 8{x} } - e \) '} over { \( x - 1 \) '} } = {"lim"} cSub { size 8{x rightarrow 1} } e rSup { size 8{x} } =e rSup { size 8{1} } =e} {}

2. limx→0ex2−cosxx2=ЛПlimx→0(ex2−cosx)'(x2)'=limx→0ex22x+sinx2x=ЛПlimx→0ex2−cosxx2=ЛПlimx→0(ex2−cosx)'(x2)'=limx→0ex22x+sinx2x=ЛП size 12{ {"lim"} cSub { size 8{x rightarrow 0} } { {e rSup { size 8{x rSup { size 6{2} } } } - "cos"x} over {x rSup {2} } } { size 24{ {}={}} } cSup {"ЛП"} { size 12{"lim"} } cSub {x rightarrow 0} { { size 12{ \( e rSup {x rSup { size 6{2} } } size 12{ - "cos"x \) '}} } over { size 12{ \( x rSup {2} size 12{ \) '}} } } size 12{ {}= {"lim"} cSub {x rightarrow 0} { { size 12{e rSup {x rSup { size 6{2} } } size 12{ cdot 2x+"sin"x}} } over { size 12{2x} } } { size 24{ {}={}} } cSup {"ЛП"} }} {}

3. limx→0ex−e−xsinx=ЛПlimx→0(ex−e−x)'(sinx)'=limx→0ex+excosx=1+11=21=2limx→0ex−e−xsinx=ЛПlimx→0(ex−e−x)'(sinx)'=limx→0ex+excosx=1+11=21=2 size 12{ {"lim"} cSub { size 8{x rightarrow 0} } { {e rSup { size 8{x} } - e rSup { size 8{ - x} } } over {"sin"x} } { size 24{ {}={}} } cSup { size 8{"ЛП"} } {"lim"} cSub { size 8{x rightarrow 0} } { { \( e rSup { size 8{x} } - e rSup { size 8{ - x} } \) '} over { \( "sin"x \) '} } = {"lim"} cSub { size 8{x rightarrow 0} } { {e rSup { size 8{x} } +e rSup { size 8{x} } } over {"cos"x} } = { {1+1} over {1} } = { {2} over {1} } =2} {}

4. limx→0esin2x−esinxx=ЛПlimx→0(esin2x−esinx)'(x)'=limx→0esin2x⋅2cos2x−esinx⋅cosx1=limx→0esin2x−esinxx=ЛПlimx→0(esin2x−esinx)'(x)'=limx→0esin2x⋅2cos2x−esinx⋅cosx1= size 12{ {"lim"} cSub { size 8{x rightarrow 0} } { {e rSup { size 8{"sin"2x} } - e rSup { size 8{"sin"x} } } over {x} } { size 24{ {}={}} } cSup { size 8{"ЛП"} } {"lim"} cSub { size 8{x rightarrow 0} } { { \( e rSup { size 8{"sin"2x} } - e rSup { size 8{"sin"x} } \) '} over { \( x \) '} } = {"lim"} cSub { size 8{x rightarrow 0} } { {e rSup { size 8{"sin"2x} } cdot 2"cos"2x - e rSup { size 8{"sin"x} } cdot "cos"x} over {1} } ={}} {}

5. limx→0eax−ebxx=ЛПlimx→0(eax−ebx)'(x)'=limx→0eax⋅a−ebx⋅b1=1⋅a−1⋅b1=a−blimx→0eax−ebxx=ЛПlimx→0(eax−ebx)'(x)'=limx→0eax⋅a−ebx⋅b1=1⋅a−1⋅b1=a−b size 12{ {"lim"} cSub { size 8{x rightarrow 0} } { {e rSup { size 8{ ital "ax"} } - e rSup { size 8{ ital "bx"} } } over {x} } { size 24{ {}={}} } cSup { size 8{"ЛП"} } {"lim"} cSub { size 8{x rightarrow 0} } { { \( e rSup { size 8{ ital "ax"} } - e rSup { size 8{ ital "bx"} } \) '} over { \( x \) '} } = {"lim"} cSub { size 8{x rightarrow 0} } { {e rSup { size 8{ ital "ax"} } cdot a - e rSup { size 8{ ital "bx"} } cdot b} over {1} } = { {1 cdot a - 1 cdot b} over {1} } =a - b} {}

6. limx→0x(e1x−1)=limx→0e1x−11x=ЛПlimx→0(e1x−1)'(1x)'limx→0e1x⋅(−1x2)(−1x2)=limx→0e1x=e1∞=e0=1limx→0x(e1x−1)=limx→0e1x−11x=ЛПlimx→0(e1x−1)'(1x)'limx→0e1x⋅(−1x2)(−1x2)=limx→0e1x=e1∞=e0=1 size 12{ {"lim"} cSub { size 8{x rightarrow 0} } x \( e rSup { size 8{ { {1} over {x} } } } - 1 \) = {"lim"} cSub { size 8{x rightarrow 0} } { {e rSup { size 8{ { {1} over {x} } } } - 1} over { { {1} over {x} } } } { size 24{ {}={}} } cSup { size 8{"ЛП"} } {"lim"} cSub { size 8{x rightarrow 0} } { { \( e rSup { size 8{ { {1} over {x} } } } - 1 \) '} over { \( { {1} over {x} } \) '} } {"lim"} cSub { size 8{x rightarrow 0} } { {e rSup { size 8{ { {1} over {x} } } } cdot \( - { {1} over {x rSup { size 8{2} } } } \) } over { \( - { {1} over {x rSup { size 8{2} } } } \) } } = {"lim"} cSub { size 8{x rightarrow 0} } e rSup { size 8{ { {1} over {x} } } } =e rSup { size 8{ { {1} over { infinity } } } } =e rSup { size 8{0} } =1} {}

7. limx→0e2x−13x=ЛПlimx→0e2x−1'3x'=limx→0e2x⋅23=2⋅e03=2⋅13=23limx→0e2x−13x=ЛПlimx→0e2x−1'3x'=limx→0e2x⋅23=2⋅e03=2⋅13=23 size 12{ {"lim"} cSub { size 8{x rightarrow 0} } { {e rSup { size 8{2x} } - 1} over {3x} } { size 24{ {}={}} } cSup { size 8{"ЛП"} } {"lim"} cSub { size 8{x rightarrow 0} } { { left (e rSup { size 8{2x} } - 1 right ) rSup { size 8{'} } } over { left (3x right ) rSup { size 8{'} } } } = {"lim"} cSub { size 8{x rightarrow 0} } { {e rSup { size 8{2x} } cdot 2} over {3} } = { {2 cdot e rSup { size 8{0} } } over {3} } = { {2 cdot 1} over {3} } = { {2} over {3} } } {}.

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This work is licensed by Beti Andonovic under a Creative Commons Attribution License (CC-BY 3.0), and is an Open Educational Resource.

Last edited by Beti Andonovic on Nov 22, 2009 3:22 pm -0600.