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Задачи од граница на функција

Module by: Beti Andonovic. E-mail the author

Summary: Solved exercises on limits of functions

Лимеси

1. limx2(3x4x1)=32421=31621=45limx2(3x4x1)=32421=31621=45 size 12{ {"lim"} cSub { size 8{x rightarrow 2} } \( 3x rSup { size 8{4} } - x - 1 \) =3 cdot 2 rSup { size 8{4} } - 2 - 1=3 cdot "16" - 2 - 1="45"} {}.

2. limx2x25x+6x2=limx2(x2)(x3)x2=limx2(x3)=23=1limx2x25x+6x2=limx2(x2)(x3)x2=limx2(x3)=23=1 size 12{ {"lim"} cSub { size 8{x rightarrow 2} } { {x rSup { size 8{2} } - 5x+6} over {x - 2} } = {"lim"} cSub { size 8{x rightarrow 2} } { { \( x - 2 \) \( x - 3 \) } over {x - 2} } = {"lim"} cSub { size 8{x rightarrow 2} } \( x - 3 \) =2 - 3= - 1} {}.

3. limx2x24x2=limx2(x2)(x+2)x2=limx2(x+2)=2+2=4limx2x24x2=limx2(x2)(x+2)x2=limx2(x+2)=2+2=4 size 12{ {"lim"} cSub { size 8{x rightarrow 2} } { {x rSup { size 8{2} } - 4} over {x - 2} } = {"lim"} cSub { size 8{x rightarrow 2} } { { \( x - 2 \) \( x+2 \) } over {x - 2} } = {"lim"} cSub { size 8{x rightarrow 2} } \( x+2 \) =2+2=4} {}.

Лимеси кадешто се крати со најголемиот степен на x

1. limx2x+13x+2=limx2x+1x3x+2x=limx2+1x3+2x=23limx2x+13x+2=limx2x+1x3x+2x=limx2+1x3+2x=23 size 12{ {"lim"} cSub { size 8{x rightarrow infinity } } { {2x+1} over {3x+2} } = {"lim"} cSub { size 8{x rightarrow infinity } } { { { {2x+1} over {x} } } over { { {3x+2} over {x} } } } = {"lim"} cSub { size 8{x rightarrow infinity } } { {2+ { {1} over {x} } } over {3+ { {2} over {x} } } } = { {2} over {3} } } {}.

2. limxx23x+12x2+x=limxx23x+1x22x2+xx2=limx13x+1x22+1x=12limxx23x+12x2+x=limxx23x+1x22x2+xx2=limx13x+1x22+1x=12 size 12{ {"lim"} cSub { size 8{x rightarrow infinity } } { {x rSup { size 8{2} } - 3x+1} over {2x rSup { size 8{2} } +x} } = {"lim"} cSub { size 8{x rightarrow infinity } } { { { {x rSup { size 8{2} } - 3x+1} over {x rSup { size 8{2} } } } } over { { {2x rSup { size 8{2} } +x} over {x rSup { size 8{2} } } } } } = {"lim"} cSub { size 8{x rightarrow infinity } } { {1 - { {3} over {x} } + { {1} over {x rSup { size 8{2} } } } } over {2+ { {1} over {x} } } } = { {1} over {2} } } {}.

3. limx3x45x2+7xx4x3+5=limx3x45x2+7xx4x4x3+5x4=limx35x2+7x311x+5x4=3limx3x45x2+7xx4x3+5=limx3x45x2+7xx4x4x3+5x4=limx35x2+7x311x+5x4=3 size 12{ {"lim"} cSub { size 8{x rightarrow infinity } } { {3x rSup { size 8{4} } - 5x rSup { size 8{2} } +7x} over {x rSup { size 8{4} } - x rSup { size 8{3} } +5} } = {"lim"} cSub { size 8{x rightarrow infinity } } { { { {3x rSup { size 8{4} } - 5x rSup { size 8{2} } +7x} over {x rSup { size 8{4} } } } } over { { {x rSup { size 8{4} } - x rSup { size 8{3} } +5} over {x rSup { size 8{4} } } } } } = {"lim"} cSub { size 8{x rightarrow infinity } } { {3 - { {5} over {x rSup { size 8{2} } } } + { {7} over {x rSup { size 8{3} } } } } over {1 - { {1} over {x} } + { {5} over {x rSup { size 8{4} } } } } } =3} {}.

4. {}limx5x2x+33x3+2x4=limx5x2x+3x33x3+2x4x3=limx5x1x2+3x33+2x24x3=03=0limx5x2x+33x3+2x4=limx5x2x+3x33x3+2x4x3=limx5x1x2+3x33+2x24x3=03=0 size 12{ {"lim"} cSub { size 8{x rightarrow infinity } } { {5x rSup { size 8{2} } - x+3} over {3x rSup { size 8{3} } +2x - 4} } = {"lim"} cSub { size 8{x rightarrow infinity } } { { { {5x rSup { size 8{2} } - x+3} over {x rSup { size 8{3} } } } } over { { {3x rSup { size 8{3} } +2x - 4} over {x rSup { size 8{3} } } } } } = {"lim"} cSub { size 8{x rightarrow infinity } } { { { {5} over {x} } - { {1} over {x rSup { size 8{2} } } } + { {3} over {x rSup { size 8{3} } } } } over {3+ { {2} over {x rSup { size 8{2} } } } - { {4} over {x rSup { size 8{3} } } } } } = { {0} over {3} } =0} {}.

5. limx6x42x3+x22x3+x23=limx6x42x3+x2x42x3+x23x4=limx62x+1x22x+1x23x4=60=limx6x42x3+x22x3+x23=limx6x42x3+x2x42x3+x23x4=limx62x+1x22x+1x23x4=60= size 12{ {"lim"} cSub { size 8{x rightarrow infinity } } { {6x rSup { size 8{4} } - 2x rSup { size 8{3} } +x rSup { size 8{2} } } over {2x rSup { size 8{3} } +x rSup { size 8{2} } - 3} } = {"lim"} cSub { size 8{x rightarrow infinity } } { { { {6x rSup { size 8{4} } - 2x rSup { size 8{3} } +x rSup { size 8{2} } } over {x rSup { size 8{4} } } } } over { { {2x rSup { size 8{3} } +x rSup { size 8{2} } - 3} over {x rSup { size 8{4} } } } } } = {"lim"} cSub { size 8{x rightarrow infinity } } { {6 - { {2} over {x} } + { {1} over {x rSup { size 8{2} } } } } over { { {2} over {x} } + { {1} over {x rSup { size 8{2} } } } - { {3} over {x rSup { size 8{4} } } } } } = { {6} over {0} } = infinity } {}.

6. limxx22xx+1=limxx22xxx+1x=limxx22xx2x+1x=limx12x1+1x=1limxx22xx+1=limxx22xxx+1x=limxx22xx2x+1x=limx12x1+1x=1 size 12{ {"lim"} cSub { size 8{x rightarrow infinity } } { { sqrt {x rSup { size 8{2} } - 2x} } over {x+1} } = {"lim"} cSub { size 8{x rightarrow infinity } } { { { { sqrt {x rSup { size 8{2} } - 2x} } over {x} } } over { { {x+1} over {x} } } } = {"lim"} cSub { size 8{x rightarrow infinity } } { { { { sqrt {x rSup { size 8{2} } - 2x} } over { sqrt {x rSup { size 8{2} } } } } } over { { {x+1} over {x} } } } = {"lim"} cSub { size 8{x rightarrow infinity } } { { sqrt {1 - { {2} over {x} } } } over {1+ { {1} over {x} } } } =1} {}.

7. limxx+x2x2x+3=limxx+x2xx2x+3x=limx1+11x2+3x=1+12=1limxx+x2x2x+3=limxx+x2xx2x+3x=limx1+11x2+3x=1+12=1 size 12{ {"lim"} cSub { size 8{x rightarrow infinity } } { {x+ sqrt {x rSup { size 8{2} } - x} } over {2x+3} } = {"lim"} cSub { size 8{x rightarrow infinity } } { { { {x+ sqrt {x rSup { size 8{2} } - x} } over {x} } } over { { {2x+3} over {x} } } } = {"lim"} cSub { size 8{x rightarrow infinity } } { {1+ sqrt {1 - { {1} over {x} } } } over {2+ { {3} over {x} } } } = { {1+1} over {2} } =1} {}.

8. limxx+3+x23x+142x4+x254=limxx+3+x23x+14x2x4+x254x=limxx+3+x23x+142x4+x254=limxx+3+x23x+14x2x4+x254x= size 12{ {"lim"} cSub { size 8{x rightarrow infinity } } { { sqrt {x+3} + nroot { size 8{4} } {x rSup { size 8{2} } - 3x+1} } over {2 sqrt {x - 4} + nroot { size 8{4} } {x rSup { size 8{2} } - 5} } } = {"lim"} cSub { size 8{x rightarrow infinity } } { { { { sqrt {x+3} + nroot { size 8{4} } {x rSup { size 8{2} } - 3x+1} } over { sqrt {x} } } } over { { {2 sqrt {x - 4} + nroot { size 8{4} } {x rSup { size 8{2} } - 5} } over { sqrt {x} } } } } ={}} {}

=limxx+3x+x23x+14x242x4x+x254x24=limx1+3x+13x+1x24214x+15x24=1+12+1=23=limxx+3x+x23x+14x242x4x+x254x24=limx1+3x+13x+1x24214x+15x24=1+12+1=23 size 12{ {}= {"lim"} cSub { size 8{x rightarrow infinity } } { { { { sqrt {x+3} } over { sqrt {x} } } + { { nroot { size 8{4} } {x rSup { size 8{2} } - 3x+1} } over { nroot { size 8{4} } {x rSup { size 8{2} } } } } } over { { {2 sqrt {x - 4} } over { sqrt {x} } } + { { nroot { size 8{4} } {x rSup { size 8{2} } - 5} } over { nroot { size 8{4} } {x rSup { size 8{2} } } } } } } = {"lim"} cSub { size 8{x rightarrow infinity } } { { sqrt {1+ { {3} over {x} } } + nroot { size 8{4} } {1 - { {3} over {x} } + { {1} over {x rSup { size 8{2} } } } } } over {2 sqrt {1 - { {4} over {x} } } + nroot { size 8{4} } {1 - { {5} over {x rSup { size 8{2} } } } } } } = { {1+1} over {2+1} } = { {2} over {3} } } {}.

9. limxx52x3+45+3x4x3+x243+x21=limxx52x3+45+3x4xx3+x243+x21x=limxx52x3+45+3x4x3+x243+x21=limxx52x3+45+3x4xx3+x243+x21x= size 12{ {"lim"} cSub { size 8{x rightarrow infinity } } { { nroot { size 8{5} } {x rSup { size 8{5} } - 2x rSup { size 8{3} } +4} +3x - 4} over { nroot { size 8{3} } {x rSup { size 8{3} } +x rSup { size 8{2} } - 4} + sqrt {x rSup { size 8{2} } - 1} } } = {"lim"} cSub { size 8{x rightarrow infinity } } { { { { nroot { size 8{5} } {x rSup { size 8{5} } - 2x rSup { size 8{3} } +4} +3x - 4} over {x} } } over { { { nroot { size 8{3} } {x rSup { size 8{3} } +x rSup { size 8{2} } - 4} + sqrt {x rSup { size 8{2} } - 1} } over {x} } } } ={}} {}=limxx52x3+45x55+3x4xx3+x243x33+x21x=limx12x2+4x55+34x1+1x4x33+11x2=1+31+1=42=2=limxx52x3+45x55+3x4xx3+x243x33+x21x=limx12x2+4x55+34x1+1x4x33+11x2=1+31+1=42=2 size 12{ {}= {"lim"} cSub { size 8{x rightarrow infinity } } { { { { nroot { size 8{5} } {x rSup { size 8{5} } - 2x rSup { size 8{3} } +4} } over { nroot { size 8{5} } {x rSup { size 8{5} } } } } + { {3x - 4} over {x} } } over { { { nroot { size 8{3} } {x rSup { size 8{3} } +x rSup { size 8{2} } - 4} } over { nroot { size 8{3} } {x rSup { size 8{3} } } } } + { { sqrt {x rSup { size 8{2} } - 1} } over {x} } } } = {"lim"} cSub { size 8{x rightarrow infinity } } { { nroot { size 8{5} } {1 - { {2} over {x rSup { size 8{2} } } } + { {4} over {x rSup { size 8{5} } } } } +3 - { {4} over {x} } } over { nroot { size 8{3} } {1+ { {1} over {x} } - { {4} over {x rSup { size 8{3} } } } } + sqrt {1 - { {1} over {x rSup { size 8{2} } } } } } } = { {1+3} over {1+1} } = { {4} over {2} } =2} {}.

10. limxx2+1+xx3+x4x=limxx2+1+xxx3+x4xx=limxx2+1x2+xx2x3+xx44xx=1+0+00+041=1limxx2+1+xx3+x4x=limxx2+1+xxx3+x4xx=limxx2+1x2+xx2x3+xx44xx=1+0+00+041=1 size 12{ {"lim"} cSub { size 8{x rightarrow infinity } } { { sqrt {x rSup { size 8{2} } +1} + sqrt {x} } over { nroot { size 8{4} } {x rSup { size 8{3} } +x} - x} } = {"lim"} cSub { size 8{x rightarrow infinity } } { { { { sqrt {x rSup { size 8{2} } +1} + sqrt {x} } over {x} } } over { { { nroot { size 8{4} } {x rSup { size 8{3} } +x} - x} over {x} } } } = {"lim"} cSub { size 8{x rightarrow infinity } } { { sqrt { { {x rSup { size 8{2} } +1} over {x rSup { size 8{2} } } } } + sqrt { { {x} over {x rSup { size 8{2} } } } } } over { nroot { size 8{4} } { { {x rSup { size 8{3} } +x} over {x rSup { size 8{4} } } } } - { {x} over {x} } } } = { { sqrt {1+0} + sqrt {0} } over { nroot { size 8{4} } {0+0} - 1} } = - 1} {}.

Лимеси кадешто се множи функцијата

1. limx(xx3)=limx((xx3)x+x3x+x3)=limx3x+x3=0limx(xx3)=limx((xx3)x+x3x+x3)=limx3x+x3=0 size 12{ {"lim"} cSub { size 8{x rightarrow infinity } } \( sqrt {x} - sqrt {x - 3} \) = {"lim"} cSub { size 8{x rightarrow infinity } } \( \( sqrt {x} - sqrt {x - 3} \) cdot { { sqrt {x} + sqrt {x - 3} } over { sqrt {x} + sqrt {x - 3} } } \) = {"lim"} cSub { size 8{x rightarrow infinity } } { {3} over { sqrt {x} + sqrt {x - 3} } } =0} {}.

2. limx((x+a)(x+b)x)=limx(((x+a)(x+b)x)(x+a)(x+b)+x(x+a)(x+b)+x)=limx((x+a)(x+b)x)=limx(((x+a)(x+b)x)(x+a)(x+b)+x(x+a)(x+b)+x)= size 12{ {"lim"} cSub { size 8{x rightarrow infinity } } \( sqrt { \( x+a \) \( x+b \) } - x \) = {"lim"} cSub { size 8{x rightarrow infinity } } \( \( sqrt { \( x+a \) \( x+b \) } - x \) cdot { { sqrt { \( x+a \) \( x+b \) } +x} over { sqrt { \( x+a \) \( x+b \) } +x} } \) ={}} {}

= lim x ( x + a ) ( x + b ) x 2 ( x + a ) ( x + b ) + x = lim x x 2 + ( a + b ) x + ab x 2 x 2 + ( a + b ) x + ab + x = = lim x ( x + a ) ( x + b ) x 2 ( x + a ) ( x + b ) + x = lim x x 2 + ( a + b ) x + ab x 2 x 2 + ( a + b ) x + ab + x = size 12{ {}= {"lim"} cSub { size 8{x rightarrow infinity } } { { \( x+a \) \( x+b \) - x rSup { size 8{2} } } over { sqrt { \( x+a \) \( x+b \) } +x} } = {"lim"} cSub { size 8{x rightarrow infinity } } { {x rSup { size 8{2} } + \( a+b \) x+ ital "ab" - x rSup { size 8{2} } } over { sqrt {x rSup { size 8{2} } + \( a+b \) x+ ital "ab"} +x} } ={}} {}
(1)
= lim x ( a + b ) x + ab x x 2 + ( a + b ) x + ab + x x = lim x ( a + b ) + ab x x 2 + ( a + b ) x + ab x 2 + x x = = lim x ( a + b ) x + ab x x 2 + ( a + b ) x + ab + x x = lim x ( a + b ) + ab x x 2 + ( a + b ) x + ab x 2 + x x = size 12{ {}= {"lim"} cSub { size 8{x rightarrow infinity } } { { { { \( a+b \) x+ ital "ab"} over {x} } } over { { { sqrt {x rSup { size 8{2} } + \( a+b \) x+ ital "ab"} +x} over {x} } } } = {"lim"} cSub { size 8{x rightarrow infinity } } { { \( a+b \) + { { ital "ab"} over {x} } } over { { { sqrt {x rSup { size 8{2} } + \( a+b \) x+ ital "ab"} } over { sqrt {x rSup { size 8{2} } } } } + { {x} over {x} } } } ={}} {}
(2)

=limx(a+b)+abx1+a+bx+abx2+1=a+b+01+0+0+1=a+b2=limx(a+b)+abx1+a+bx+abx2+1=a+b+01+0+0+1=a+b2 size 12{ {}= {"lim"} cSub { size 8{x rightarrow infinity } } { { \( a+b \) + { { ital "ab"} over {x} } } over { sqrt {1+ { {a+b} over {x} } + { { ital "ab"} over {x rSup { size 8{2} } } } } +1} } = { {a+b+0} over { sqrt {1+0+0} +1} } = { {a+b} over {2} } } {}.

3. limx((x+1)23(x1)23)=limx((x+1)23(x1)23)= size 12{ {"lim"} cSub { size 8{x rightarrow infinity } } \( nroot { size 8{3} } { \( x+1 \) rSup { size 8{2} } } - nroot { size 8{3} } { \( x - 1 \) rSup { size 8{2} } } \) ={}} {}

= lim x ( x + 1 ) 2 3 ( x 1 ) 2 3 ( x + 1 ) 4 3 + ( x + 1 ) 2 ( x 1 ) 2 3 + ( x 1 ) 4 3 ( x + 1 ) 4 3 + ( x + 1 ) 2 ( x 1 ) 2 3 + ( x 1 ) 4 3 = = lim x ( x + 1 ) 2 3 ( x 1 ) 2 3 ( x + 1 ) 4 3 + ( x + 1 ) 2 ( x 1 ) 2 3 + ( x 1 ) 4 3 ( x + 1 ) 4 3 + ( x + 1 ) 2 ( x 1 ) 2 3 + ( x 1 ) 4 3 = size 12{ {}= {"lim"} cSub { size 8{x rightarrow infinity } } left ( left ( nroot { size 8{3} } { \( x+1 \) rSup { size 8{2} } } - nroot { size 8{3} } { \( x - 1 \) rSup { size 8{2} } } right ) cdot { { nroot { size 8{3} } { \( x+1 \) rSup { size 8{4} } } + nroot { size 8{3} } { \( x+1 \) rSup { size 8{2} } \( x - 1 \) rSup { size 8{2} } } + nroot { size 8{3} } { \( x - 1 \) rSup { size 8{4} } } } over { nroot { size 8{3} } { \( x+1 \) rSup { size 8{4} } } + nroot { size 8{3} } { \( x+1 \) rSup { size 8{2} } \( x - 1 \) rSup { size 8{2} } } + nroot { size 8{3} } { \( x - 1 \) rSup { size 8{4} } } } } right )={}} {}
(3)
= lim x ( x + 1 ) 2 ( x 1 ) 2 x 4 + 4x 3 + 6x 2 + 4x + 1 3 + x 4 2x + 1 3 + x 4 4x 3 + 6x 2 4x + 1 3 = = lim x ( x + 1 ) 2 ( x 1 ) 2 x 4 + 4x 3 + 6x 2 + 4x + 1 3 + x 4 2x + 1 3 + x 4 4x 3 + 6x 2 4x + 1 3 = size 12{ {}= {"lim"} cSub { size 8{x rightarrow infinity } } left ( { { \( x+1 \) rSup { size 8{2} } - \( x - 1 \) rSup { size 8{2} } } over { nroot { size 8{3} } {x rSup { size 8{4} } +4x rSup { size 8{3} } +6x rSup { size 8{2} } +4x+1} + nroot { size 8{3} } {x rSup { size 8{4} } - 2x+1} + nroot { size 8{3} } {x rSup { size 8{4} } - 4x rSup { size 8{3} } +6x rSup { size 8{2} } - 4x+1} } } right )={}} {}
(4)
= lim x 4x x 4 + 4x 3 + 6x 2 + 4x + 1 3 + x 4 2x + 1 3 + x 4 4x 3 + 6x 2 4x + 1 3 = = lim x 4x x 4 + 4x 3 + 6x 2 + 4x + 1 3 + x 4 2x + 1 3 + x 4 4x 3 + 6x 2 4x + 1 3 = size 12{ {}= {"lim"} cSub { size 8{x rightarrow infinity } } left ( { {4x} over { nroot { size 8{3} } {x rSup { size 8{4} } +4x rSup { size 8{3} } +6x rSup { size 8{2} } +4x+1} + nroot { size 8{3} } {x rSup { size 8{4} } - 2x+1} + nroot { size 8{3} } {x rSup { size 8{4} } - 4x rSup { size 8{3} } +6x rSup { size 8{2} } - 4x+1} } } right )={}} {}
(5)

= lim x 4x x 4 + 4x 3 + 6x 2 + 4x + 1 3 + x 4 2x + 1 3 + x 4 4x 3 + 6x 2 4x + 1 3 x 4 3 x 4 3 = = lim x 4x x 4 + 4x 3 + 6x 2 + 4x + 1 3 + x 4 2x + 1 3 + x 4 4x 3 + 6x 2 4x + 1 3 x 4 3 x 4 3 = size 12{ {}= {"lim"} cSub { size 8{x rightarrow infinity } } left ( { {4x} over { nroot { size 8{3} } {x rSup { size 8{4} } +4x rSup { size 8{3} } +6x rSup { size 8{2} } +4x+1} + nroot { size 8{3} } {x rSup { size 8{4} } - 2x+1} + nroot { size 8{3} } {x rSup { size 8{4} } - 4x rSup { size 8{3} } +6x rSup { size 8{2} } - 4x+1} } } cdot { { nroot { size 8{3} } {x rSup { size 8{4} } } } over { nroot { size 8{3} } {x rSup { size 8{4} } } } } right )={}} {}

=01+0+0+0+03+10+03+10+00+03+=03=0=01+0+0+0+03+10+03+10+00+03+=03=0 size 12{ {}= { {0} over { nroot { size 8{3} } {1+0+0+0+0} + nroot { size 8{3} } {1 - 0+0} + nroot { size 8{3} } {1 - 0+0 - 0+0} +{}} } = { {0} over {3} } =0} {}.

4. limx01+x2312x4x+x2=limx01+x2312x4+11x+x2=limx01+x2312x4x+x2=limx01+x2312x4+11x+x2= size 12{ {"lim"} cSub { size 8{x rightarrow 0} } { { nroot { size 8{3} } {1+x rSup { size 8{2} } } - nroot { size 8{4} } {1 - 2x} } over {x+x rSup { size 8{2} } } } = {"lim"} cSub { size 8{x rightarrow 0} } { { nroot { size 8{3} } {1+x rSup { size 8{2} } } - nroot { size 8{4} } {1 - 2x} +1 - 1} over {x+x rSup { size 8{2} } } } ={}} {}

= lim x 0 1 + x 2 3 1 ( 1 2x 4 1 ) x + x 2 = lim x 0 1 + x 2 3 1 x + x 2 lim x 0 ( 1 2x 4 1 ) x + x 2 = = lim x 0 1 + x 2 3 1 ( 1 2x 4 1 ) x + x 2 = lim x 0 1 + x 2 3 1 x + x 2 lim x 0 ( 1 2x 4 1 ) x + x 2 = size 12{ {}= {"lim"} cSub { size 8{x rightarrow 0} } { { nroot { size 8{3} } {1+x rSup { size 8{2} } } - 1 - \( nroot { size 8{4} } {1 - 2x} - 1 \) } over {x+x rSup { size 8{2} } } } = {"lim"} cSub { size 8{x rightarrow 0} } { { nroot { size 8{3} } {1+x rSup { size 8{2} } } - 1} over {x+x rSup { size 8{2} } } } - {"lim"} cSub { size 8{x rightarrow 0} } { { \( nroot { size 8{4} } {1 - 2x} - 1 \) } over {x+x rSup { size 8{2} } } } ={}} {}
(6)
= lim x 0 ( 1 + x 2 3 1 ) ( ( 1 + x 2 ) 2 3 + 1 + x 2 3 + 1 ) ( x + x 2 ) ( ( 1 + x 2 ) 2 3 + 1 + x 2 3 + 1 ) = lim x 0 ( 1 + x 2 3 1 ) ( ( 1 + x 2 ) 2 3 + 1 + x 2 3 + 1 ) ( x + x 2 ) ( ( 1 + x 2 ) 2 3 + 1 + x 2 3 + 1 ) size 12{ {}= {"lim"} cSub { size 8{x rightarrow 0} } { { \( nroot { size 8{3} } {1+x rSup { size 8{2} } } - 1 \) \( nroot { size 8{3} } { \( 1+x rSup { size 8{2} } \) rSup { size 8{2} } } + nroot { size 8{3} } {1+x rSup { size 8{2} } } +1 \) } over { \( x+x rSup { size 8{2} } \) \( nroot { size 8{3} } { \( 1+x rSup { size 8{2} } \) rSup { size 8{2} } } + nroot { size 8{3} } {1+x rSup { size 8{2} } } +1 \) } } - {}} {}
(7)
lim x 0 ( 1 2x 4 1 ) ( ( 1 2x ) 3 4 + ( 1 2x ) 2 4 + ( 1 2x ) 4 + 1 ) ( x + x 2 ) ( ( 1 2x ) 3 4 + ( 1 2x ) 2 4 + ( 1 2x ) 4 + 1 ) = lim x 0 ( 1 2x 4 1 ) ( ( 1 2x ) 3 4 + ( 1 2x ) 2 4 + ( 1 2x ) 4 + 1 ) ( x + x 2 ) ( ( 1 2x ) 3 4 + ( 1 2x ) 2 4 + ( 1 2x ) 4 + 1 ) = size 12{ - {"lim"} cSub { size 8{x rightarrow 0} } { { \( nroot { size 8{4} } {1 - 2x} - 1 \) \( nroot { size 8{4} } { \( 1 - 2x \) rSup { size 8{3} } } + nroot { size 8{4} } { \( 1 - 2x \) rSup { size 8{2} } } + nroot { size 8{4} } { \( 1 - 2x \) } +1 \) } over { \( x+x rSup { size 8{2} } \) \( nroot { size 8{4} } { \( 1 - 2x \) rSup { size 8{3} } } + nroot { size 8{4} } { \( 1 - 2x \) rSup { size 8{2} } } + nroot { size 8{4} } { \( 1 - 2x \) } +1 \) } } ={}} {}
(8)
= lim x 0 ( 1 + x 2 3 ) 3 1 3 x ( 1 + x ) ( ( 1 + x 2 ) 2 3 + 1 + x 2 3 + 1 ) = lim x 0 ( 1 + x 2 3 ) 3 1 3 x ( 1 + x ) ( ( 1 + x 2 ) 2 3 + 1 + x 2 3 + 1 ) size 12{ {}= {"lim"} cSub { size 8{x rightarrow 0} } { { \( nroot { size 8{3} } {1+x rSup { size 8{2} } } \) rSup { size 8{3} } - 1 rSup { size 8{3} } } over {x \( 1+x \) \( nroot { size 8{3} } { \( 1+x rSup { size 8{2} } \) rSup { size 8{2} } } + nroot { size 8{3} } {1+x rSup { size 8{2} } } +1 \) } } - {}} {}
(9)
lim x 0 ( 1 2x 4 ) 4 1 4 x ( 1 + x ) ( ( 1 2x ) 3 4 + ( 1 2x ) 2 4 + ( 1 2x ) 4 + 1 ) = lim x 0 ( 1 2x 4 ) 4 1 4 x ( 1 + x ) ( ( 1 2x ) 3 4 + ( 1 2x ) 2 4 + ( 1 2x ) 4 + 1 ) = size 12{ - {"lim"} cSub { size 8{x rightarrow 0} } { { \( nroot { size 8{4} } {1 - 2x} \) rSup { size 8{4} } - 1 rSup { size 8{4} } } over {x \( 1+x \) \( nroot { size 8{4} } { \( 1 - 2x \) rSup { size 8{3} } } + nroot { size 8{4} } { \( 1 - 2x \) rSup { size 8{2} } } + nroot { size 8{4} } { \( 1 - 2x \) } +1 \) } } ={}} {}
(10)
= lim x 0 1 + x 2 1 x ( 1 + x ) ( ( 1 + x 2 ) 2 3 + 1 + x 2 3 + 1 ) = lim x 0 1 + x 2 1 x ( 1 + x ) ( ( 1 + x 2 ) 2 3 + 1 + x 2 3 + 1 ) size 12{ {}= {"lim"} cSub { size 8{x rightarrow 0} } { {1+x rSup { size 8{2} } - 1} over {x \( 1+x \) \( nroot { size 8{3} } { \( 1+x rSup { size 8{2} } \) rSup { size 8{2} } } + nroot { size 8{3} } {1+x rSup { size 8{2} } } +1 \) } } - {}} {}
(11)
lim x 0 1 2x 1 x ( 1 + x ) ( ( 1 2x ) 3 4 + ( 1 2x ) 2 4 + ( 1 2x ) 4 + 1 ) = lim x 0 1 2x 1 x ( 1 + x ) ( ( 1 2x ) 3 4 + ( 1 2x ) 2 4 + ( 1 2x ) 4 + 1 ) = size 12{ - {"lim"} cSub { size 8{x rightarrow 0} } { {1 - 2x - 1} over {x \( 1+x \) \( nroot { size 8{4} } { \( 1 - 2x \) rSup { size 8{3} } } + nroot { size 8{4} } { \( 1 - 2x \) rSup { size 8{2} } } + nroot { size 8{4} } { \( 1 - 2x \) } +1 \) } } ={}} {}
(12)
= lim x 0 x ( 1 + x ) ( ( 1 + x 2 ) 2 3 + 1 + x 2 3 + 1 ) = lim x 0 x ( 1 + x ) ( ( 1 + x 2 ) 2 3 + 1 + x 2 3 + 1 ) size 12{ {}= {"lim"} cSub { size 8{x rightarrow 0} } { {x} over { \( 1+x \) \( nroot { size 8{3} } { \( 1+x rSup { size 8{2} } \) rSup { size 8{2} } } + nroot { size 8{3} } {1+x rSup { size 8{2} } } +1 \) } } - {}} {}
(13)
lim x 0 2 ( 1 + x ) ( ( 1 2x ) 3 4 + ( 1 2x ) 2 4 + ( 1 2x ) 4 + 1 ) = lim x 0 2 ( 1 + x ) ( ( 1 2x ) 3 4 + ( 1 2x ) 2 4 + ( 1 2x ) 4 + 1 ) = size 12{ - {"lim"} cSub { size 8{x rightarrow 0} } { { - 2} over { \( 1+x \) \( nroot { size 8{4} } { \( 1 - 2x \) rSup { size 8{3} } } + nroot { size 8{4} } { \( 1 - 2x \) rSup { size 8{2} } } + nroot { size 8{4} } { \( 1 - 2x \) } +1 \) } } ={}} {}
(14)
= 0 ( 1 + 0 ) ( ( 1 + 0 2 ) 2 3 + 1 + 0 2 3 + 1 ) + 2 ( 1 + 0 ) ( ( 1 0 ) 3 4 + ( 1 0 ) 2 4 + ( 1 0 ) 4 + 1 ) = = 0 ( 1 + 0 ) ( ( 1 + 0 2 ) 2 3 + 1 + 0 2 3 + 1 ) + 2 ( 1 + 0 ) ( ( 1 0 ) 3 4 + ( 1 0 ) 2 4 + ( 1 0 ) 4 + 1 ) = size 12{ {}= { {0} over { \( 1+0 \) \( nroot { size 8{3} } { \( 1+0 rSup { size 8{2} } \) rSup { size 8{2} } } + nroot { size 8{3} } {1+0 rSup { size 8{2} } } +1 \) } } + { {2} over { \( 1+0 \) \( nroot { size 8{4} } { \( 1 - 0 \) rSup { size 8{3} } } + nroot { size 8{4} } { \( 1 - 0 \) rSup { size 8{2} } } + nroot { size 8{4} } { \( 1 - 0 \) } +1 \) } } ={}} {}
(15)

=01(1+1+1)+21(1+1+1+1)=0+12=12=01(1+1+1)+21(1+1+1+1)=0+12=12 size 12{ {}= { {0} over {1 cdot \( 1+1+1 \) } } + { {2} over {1 cdot \( 1+1+1+1 \) } } =0+ { {1} over {2} } = { {1} over {2} } } {}.

Лимеси со бројот е

1. limxx+3xx=limx1+3xx=limx1+1x3x=limx1+1x3x33=e3limxx+3xx=limx1+3xx=limx1+1x3x=limx1+1x3x33=e3 size 12{ {"lim"} cSub { size 8{x rightarrow infinity } } left ( { {x+3} over {x} } right ) rSup { size 8{x} } = {"lim"} cSub { size 8{x rightarrow infinity } } left (1+ { {3} over {x} } right ) rSup { size 8{x} } = {"lim"} cSub { size 8{x rightarrow infinity } } left (1+ { {1} over { { {x} over {3} } } } right ) rSup { size 8{x} } = {"lim"} cSub { size 8{x rightarrow infinity } } left (1+ { {1} over { { {x} over {3} } } } right ) rSup { size 8{ { {x} over {3} } cdot 3} } =e rSup { size 8{3} } } {}

2. limxxx+4x=limx1x+4xx=1limxx+4xx=1e4=e4limxxx+4x=limx1x+4xx=1limxx+4xx=1e4=e4 size 12{ {"lim"} cSub { size 8{x rightarrow infinity } } left ( { {x} over {x+4} } right ) rSup { size 8{x} } = {"lim"} cSub { size 8{x rightarrow infinity } } { {1} over { left ( { {x+4} over {x} } right ) rSup { size 8{x} } } } = { {1} over { {"lim"} cSub { size 8{x rightarrow infinity } } left ( { {x+4} over {x} } right ) rSup { size 8{x} } } } = { {1} over {e rSup { size 8{4} } } } =e rSup { size 8{ - 4} } } {}

3. limxx3x+1x=limx1+x3x+11x=limx1+4x+1x=limxx3x+1x=limx1+x3x+11x=limx1+4x+1x= size 12{ {"lim"} cSub { size 8{x rightarrow infinity } } left ( { {x - 3} over {x+1} } right ) rSup { size 8{x} } = {"lim"} cSub { size 8{x rightarrow infinity } } left (1+ { {x - 3} over {x+1} } - 1 right ) rSup { size 8{x} } = {"lim"} cSub { size 8{x rightarrow infinity } } left (1+ { { - 4} over {x+1} } right ) rSup { size 8{x} } ={}} {}

= lim x 1 + 1 x + 1 4 x + 1 4 4x x + 1 = e lim x 4x x + 1 = e 4 = lim x 1 + 1 x + 1 4 x + 1 4 4x x + 1 = e lim x 4x x + 1 = e 4 size 12{ {}= {"lim"} cSub { size 8{x rightarrow infinity } } left (1+ { {1} over { { {x+1} over { - 4} } } } right ) rSup { size 8{ { {x+1} over { - 4} } left ( { { - 4x} over {x+1} } right )} } =e rSup { size 8{ {"lim"} cSub { size 6{x rightarrow infinity } } { { - 4x} over {x+1} } } } =e rSup { - 4} } {}

= 1 1 3x + 2 6 x + 1 3 3x + 2 6 6 3x + 2 = lim x 1 1 3x + 2 6 3x + 2 6 x + 1 3 6 3x + 2 = e 1 lim x 6x + 6 9x + 6 = = 1 1 3x + 2 6 x + 1 3 3x + 2 6 6 3x + 2 = lim x 1 1 3x + 2 6 3x + 2 6 x + 1 3 6 3x + 2 = e 1 lim x 6x + 6 9x + 6 = size 12{ {}= left (1 - { {1} over { { {3x+2} over {6} } } } right ) rSup { size 8{ { {x+1} over {3} } cdot { {3x+2} over {6} } cdot { {6} over {3x+2} } } } = {"lim"} cSub { size 8{x - > infinity } } left [ left (1 - { {1} over { { {3x+2} over {6} } } } right ) rSup { size 8{ { {3x+2} over {6} } } } right ] rSup { size 8{ { {x+1} over {3} } cdot { {6} over {3x+2} } } } = left (e rSup { size 8{ - 1} } right ) rSup { size 8{ {"lim"} cSub { size 6{x - > infinity } } { {6x+6} over {9x+6} } } } ={}} {} = e 1 lim x 6x x + 6 x 9x x + 6 x = e 1 2 3 = e 2 3 = e 1 lim x 6x x + 6 x 9x x + 6 x = e 1 2 3 = e 2 3 size 12{ {}= left (e rSup { size 8{ - 1} } right ) rSup { size 8{ {"lim"} cSub { size 6{x - > infinity } } { { { {6x} over {x} } + { {6} over {x} } } over { { {9x} over {x} } + { {6} over {x} } } } } } = left (e rSup { - 1} right ) rSup { { {2} over {3} } } size 12{ {}=e rSup { - { {2} over {3} } } }} {}

5. limxx2+1x21x2=limx1+x2+1x211x2=limx1+x2+1x2+1x21x2=limxx2+1x21x2=limx1+x2+1x211x2=limx1+x2+1x2+1x21x2= size 12{ {"lim"} cSub { size 8{x - > infinity } } left ( { {x rSup { size 8{2} } +1} over {x rSup { size 8{2} } - 1} } right ) rSup { size 8{x rSup { size 6{2} } } } = {"lim"} cSub {x - > infinity } left ( size 12{1+ { {x rSup {2} size 12{+1}} over {x rSup {2} size 12{ - 1}} } - 1} right ) rSup {x rSup { size 6{2} } } size 12{ {}= {"lim"} cSub {x - > infinity } left ( size 12{1+ { {x rSup {2} size 12{+1 - x rSup {2} } size 12{+1}} over {x rSup {2} size 12{ - 1}} } } right ) rSup {x rSup { size 6{2} } } } size 12{ {}={}}} {}

= lim x 1 + 1 x 2 1 2 x 2 x 2 1 2 2 x 2 1 = 1 + 1 x 2 1 2 x 2 1 2 2x 2 x 2 1 = e lim x 2x 2 x 2 1 = e lim x 2x 2 x 2 x 2 x 2 1 x 2 = e 2 = lim x 1 + 1 x 2 1 2 x 2 x 2 1 2 2 x 2 1 = 1 + 1 x 2 1 2 x 2 1 2 2x 2 x 2 1 = e lim x 2x 2 x 2 1 = e lim x 2x 2 x 2 x 2 x 2 1 x 2 = e 2 size 12{ {}= {"lim"} cSub { size 8{x - > infinity } } left (1+ { {1} over { { {x rSup { size 8{2} } - 1} over {2} } } } right ) rSup { size 8{x rSup { size 6{2} } cdot { {x rSup { size 6{2} } - 1} over {2} } cdot { {2} over {x rSup { size 6{2} } - 1} } } } = left [ left (1+ { {1} over { { {x rSup {2} size 12{ - 1}} over {2} } } } right ) rSup { { {x rSup { size 6{2} } - 1} over {2} } } right ] rSup { { {2x rSup { size 6{2} } } over {x rSup { size 6{2} } - 1} } } size 12{ {}=e rSup { {"lim"} cSub { size 6{x - > infinity } } { {2x rSup { size 6{2} } } over {x rSup { size 6{2} } - 1} } } } size 12{ {}=e rSup { {"lim"} cSub { size 6{x - > infinity } } { { { {2x rSup { size 6{2} } } over {x rSup { size 6{2} } } } } over { { {x rSup { size 6{2} } } over {x rSup { size 6{2} } } } - { {1} over {x rSup { size 6{2} } } } } } } } size 12{ {}=e rSup {2} }} {}

6. limx1+1x2x=limx1+1x2xx21x2=limx1+1x2x2xx2=elimxxx2=limx1+1x2x=limx1+1x2xx21x2=limx1+1x2x2xx2=elimxxx2= size 12{ {"lim"} cSub { size 8{x - > infinity } } left (1+ { {1} over {x rSup { size 8{2} } } } right ) rSup { size 8{x} } = {"lim"} cSub { size 8{x - > infinity } } left (1+ { {1} over {x rSup { size 8{2} } } } right ) rSup { size 8{x cdot x rSup { size 6{2} } cdot { {1} over {x rSup { size 6{2} } } } } } = {"lim"} cSub {x - > infinity } left [ size 12{ left (1+ { {1} over {x rSup {2} } } right ) rSup {x rSup { size 6{2} } } } right ] rSup { { {x} over {x rSup { size 6{2} } } } } size 12{ {}=e rSup { {"lim"} cSub { size 6{x - > infinity } } { {x} over {x rSup { size 6{2} } } } } } size 12{ {}={}}} {}

=elimxxx2x2x2=e0=1=elimxxx2x2x2=e0=1 size 12{ {}=e rSup { size 8{ {"lim"} cSub { size 6{x - > infinity } } { { { {x} over {x rSup { size 6{2} } } } } over { { {x rSup { size 6{2} } } over {x rSup { size 6{2} } } } } } } } =e rSup {0} size 12{ {}=1}} {}.

7. limxx22x+1x24x+2x=limx1+x22x+1x24x+21x=limx1+x22x+1x2+4x2x24x+2x=limxx22x+1x24x+2x=limx1+x22x+1x24x+21x=limx1+x22x+1x2+4x2x24x+2x= size 12{ {"lim"} cSub { size 8{x - > infinity } } left ( { {x rSup { size 8{2} } - 2x+1} over {x rSup { size 8{2} } - 4x+2} } right ) rSup { size 8{x} } = {"lim"} cSub { size 8{x - > infinity } } left (1+ { {x rSup { size 8{2} } - 2x+1} over {x rSup { size 8{2} } - 4x+2} } - 1 right ) rSup { size 8{x} } = {"lim"} cSub { size 8{x - > infinity } } left (1+ { {x rSup { size 8{2} } - 2x+1 - x rSup { size 8{2} } +4x - 2} over {x rSup { size 8{2} } - 4x+2} } right ) rSup { size 8{x} } ={}} {}

= lim x 1 + 2x 1 x 2 4x + 2 x = lim x 1 + 1 x 2 4x + 2 2x 1 x x 2 4x + 2 2x 1 2x 1 x 2 4x + 2 = lim x 1 + 1 x 2 4x + 2 2x 1 x 2 4x + 2 2x 1 2x 2 x x 2 4x + 2 = = lim x 1 + 2x 1 x 2 4x + 2 x = lim x 1 + 1 x 2 4x + 2 2x 1 x x 2 4x + 2 2x 1 2x 1 x 2 4x + 2 = lim x 1 + 1 x 2 4x + 2 2x 1 x 2 4x + 2 2x 1 2x 2 x x 2 4x + 2 = size 12{ {}= {"lim"} cSub { size 8{x - > infinity } } left (1+ { {2x - 1} over {x rSup { size 8{2} } - 4x+2} } right ) rSup { size 8{x} } = {"lim"} cSub { size 8{x - > infinity } } left (1+ { {1} over { { {x rSup { size 8{2} } - 4x+2} over {2x - 1} } } } right ) rSup { size 8{x cdot { {x rSup { size 6{2} } - 4x+2} over {2x - 1} } cdot { {2x - 1} over {x rSup { size 6{2} } - 4x+2} } } } = {"lim"} cSub {x - > infinity } left [ size 12{ left (1+ { {1} over { { {x rSup {2} size 12{ - 4x+2}} over {2x - 1} } } } right ) rSup { { {x rSup { size 6{2} } - 4x+2} over {2x - 1} } } } right ] rSup { { {2x rSup { size 6{2} } - x} over {x rSup { size 6{2} } - 4x+2} } } size 12{ {}={}}} {} = e lim x 2x 2 x 2 x x 2 x 2 x 2 4x x 2 + 2 x 2 = e 2 = e lim x 2x 2 x 2 x x 2 x 2 x 2 4x x 2 + 2 x 2 = e 2 size 12{ {}=e rSup { size 8{ {"lim"} cSub { size 6{x - > infinity } } { { { {2x rSup { size 6{2} } } over {x rSup { size 6{2} } } } - { {x} over {x rSup { size 6{2} } } } } over { { {x rSup { size 6{2} } } over {x rSup { size 6{2} } } } - { {4x} over {x rSup { size 6{2} } } } + { {2} over {x rSup { size 6{2} } } } } } } } =e rSup {2} } {}

9. limx01+tg2x12x=limx01+tg2x12xtg2z1tg2x=limx01+tg2x1tg2xtg2x2x=elimx0tg2x2x2=limx01+tg2x12x=limx01+tg2x12xtg2z1tg2x=limx01+tg2x1tg2xtg2x2x=elimx0tg2x2x2= size 12{ {"lim"} cSub { size 8{x rightarrow 0} } left (1+ ital "tg" rSup { size 8{2} } sqrt {x} right ) rSup { size 8{ { {1} over {2x} } } } = {"lim"} cSub { size 8{x rightarrow 0} } left (1+ ital "tg" rSup { size 8{2} } sqrt {x} right ) rSup { size 8{ { {1} over {2x} } cdot ital "tg" rSup { size 6{2} } sqrt {z} cdot { {1} over { ital "tg" rSup { size 6{2} } sqrt {x} } } } } = {"lim"} cSub {x rightarrow 0} left [ size 12{ left (1+ ital "tg" rSup {2} sqrt { size 12{x} } right ) rSup { { {1} over { ital "tg" rSup { size 6{2} } sqrt {x} } } } } right ] rSup { { { ital "tg" rSup { size 6{2} } sqrt {x} } over {2x} } } size 12{ {}=e rSup { {"lim"} cSub { size 6{x rightarrow 0} } { { ital "tg" rSup { size 6{2} } sqrt {x} } over {2 left ( sqrt {x} right ) rSup { size 6{2} } } } } } size 12{ {}={}}} {}

= e 1 2 lim x 0 tg 2 x x 2 = e 1 2 1 2 = e 1 2 = e 1 2 lim x 0 tg 2 x x 2 = e 1 2 1 2 = e 1 2 size 12{ {}=e rSup { size 8{ { {1} over {2} } {"lim"} cSub { size 6{x rightarrow 0} } left ( { { ital "tg" rSup { size 6{2} } sqrt {x} } over { sqrt {x} } } right ) rSup { size 6{2} } } } =e rSup { { {1} over {2} } cdot 1 rSup { size 6{2} } } size 12{ {}=e rSup { { {1} over {2} } } }} {}

= e 1 2 lim x 0 tg 2 x x 2 = e 1 2 1 2 = e 1 2 = e 1 2 lim x 0 tg 2 x x 2 = e 1 2 1 2 = e 1 2 size 12{ {}=e rSup { size 8{ { {1} over {2} } {"lim"} cSub { size 6{x rightarrow 0} } left ( { { ital "tg" rSup { size 6{2} } sqrt {x} } over { sqrt {x} } } right ) rSup { size 6{2} } } } =e rSup { { {1} over {2} } cdot 1 rSup { size 6{2} } } size 12{ {}=e rSup { { {1} over {2} } } }} {}
(16)

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