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Tentukanlah x→2lim​x2−4x+41−cos(x−2)​​

Pertanyaan

Tentukanlah limit as x rightwards arrow 2 of fraction numerator 1 minus square root of cos left parenthesis x minus 2 right parenthesis end root over denominator x squared minus 4 x plus 4 end fraction

N. Puspita

Master Teacher

Jawaban terverifikasi

Jawaban

 limit as x rightwards arrow 2 of fraction numerator 1 minus square root of cos left parenthesis x minus 2 right parenthesis end root over denominator x squared minus 4 x plus 4 end fraction equals 1 fourth.

Pembahasan

Ingat kembali beberapa sifat berikut.

  • limit as x rightwards arrow c of fraction numerator sin space a left parenthesis x minus c right parenthesis over denominator b left parenthesis x minus c right parenthesis end fraction equals limit as x rightwards arrow 0 of fraction numerator space a left parenthesis x minus c right parenthesis over denominator sin space b left parenthesis x minus c right parenthesis end fraction equals a over b

 

  • left parenthesis a plus b right parenthesis left parenthesis a minus b right parenthesis equals a squared minus b squared 

 

  • table attributes columnalign right center left columnspacing 0px end attributes row cell cos space 2 x end cell equals cell 1 minus 2 space sin squared space x end cell row cell cos space x end cell equals cell 1 minus 2 space sin squared space 1 half x end cell end table 

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  • limit as x rightwards arrow c of k space f left parenthesis x right parenthesis equals k space limit as x rightwards arrow c of space f left parenthesis x right parenthesis

 

  • limit as x rightwards arrow c of f left parenthesis x right parenthesis times g left parenthesis x right parenthesis equals space limit as x rightwards arrow c of space f left parenthesis x right parenthesis times limit as x rightwards arrow c of g left parenthesis x right parenthesis 

Dengan aturan di atas diperoleh.

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as x rightwards arrow 2 of fraction numerator 1 minus square root of cos left parenthesis x minus 2 right parenthesis end root over denominator x squared minus 4 x plus 4 end fraction end cell row blank equals cell limit as x rightwards arrow 2 of fraction numerator 1 minus square root of cos left parenthesis x minus 2 right parenthesis end root over denominator x squared minus 4 x plus 4 end fraction times fraction numerator 1 plus square root of cos left parenthesis x minus 2 right parenthesis end root over denominator 1 plus square root of cos left parenthesis x minus 2 right parenthesis end root end fraction end cell row blank equals cell limit as x rightwards arrow 2 of fraction numerator 1 minus cos left parenthesis x minus 2 right parenthesis over denominator left parenthesis x squared minus 4 x plus 4 right parenthesis left parenthesis 1 plus square root of cos left parenthesis x minus 2 right parenthesis end root right parenthesis end fraction end cell row blank equals cell limit as x rightwards arrow 2 of fraction numerator 1 minus open parentheses 1 minus 2 sin squared begin display style 1 half end style left parenthesis x minus 2 right parenthesis close parentheses over denominator left parenthesis x minus 2 right parenthesis left parenthesis x minus 2 right parenthesis left parenthesis 1 plus square root of cos left parenthesis x minus 2 right parenthesis end root right parenthesis end fraction end cell row blank equals cell limit as x rightwards arrow 2 of fraction numerator 1 minus 1 plus 2 sin squared 1 half left parenthesis x minus 2 right parenthesis over denominator left parenthesis x minus 2 right parenthesis left parenthesis x minus 2 right parenthesis left parenthesis 1 plus square root of cos left parenthesis x minus 2 right parenthesis end root right parenthesis end fraction end cell row blank equals cell limit as x rightwards arrow 2 of fraction numerator 2 sin squared 1 half left parenthesis x minus 2 right parenthesis over denominator left parenthesis x minus 2 right parenthesis left parenthesis x minus 2 right parenthesis left parenthesis 1 plus square root of cos left parenthesis x minus 2 right parenthesis end root right parenthesis end fraction end cell row blank equals cell limit as x rightwards arrow 2 of fraction numerator 2 sin squared 1 half left parenthesis x minus 2 right parenthesis over denominator left parenthesis x minus 2 right parenthesis left parenthesis x minus 2 right parenthesis left parenthesis 1 plus square root of cos left parenthesis x minus 2 right parenthesis end root right parenthesis end fraction end cell row blank equals cell limit as x rightwards arrow 2 of fraction numerator 2 sin 1 half left parenthesis x minus 2 right parenthesis times sin 1 half left parenthesis x minus 2 right parenthesis over denominator left parenthesis x minus 2 right parenthesis left parenthesis begin display style x minus 2 end style right parenthesis left parenthesis 1 plus square root of cos left parenthesis x minus 2 right parenthesis end root right parenthesis end fraction end cell row blank equals cell 2 times limit as x rightwards arrow 2 of fraction numerator sin 1 half left parenthesis x minus 2 right parenthesis over denominator left parenthesis x minus 2 right parenthesis end fraction times limit as x rightwards arrow 2 of fraction numerator sin 1 half left parenthesis x minus 2 right parenthesis over denominator left parenthesis x minus 2 right parenthesis end fraction times limit as x rightwards arrow 2 of fraction numerator 1 over denominator left parenthesis 1 plus square root of cos left parenthesis x minus 2 right parenthesis end root right parenthesis end fraction end cell row blank equals cell 2 times 1 half times 1 half times fraction numerator 1 over denominator 1 plus square root of cos left parenthesis 2 minus 2 right parenthesis end root end fraction end cell row blank equals cell 1 half times fraction numerator 1 over denominator 1 plus square root of cos left parenthesis 0 right parenthesis end root end fraction end cell row blank equals cell 1 half times fraction numerator 1 over denominator 1 plus square root of 1 end fraction end cell row blank equals cell 1 half times fraction numerator 1 over denominator 1 plus 1 end fraction end cell row blank equals cell 1 half times 1 half end cell row blank equals cell 1 fourth end cell end table  

Jadi, limit as x rightwards arrow 2 of fraction numerator 1 minus square root of cos left parenthesis x minus 2 right parenthesis end root over denominator x squared minus 4 x plus 4 end fraction equals 1 fourth.

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