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x→1lim​(x−1)tan(3x−3)cos(2x−2)−1​=...

Pertanyaan

limit as x rightwards arrow 1 of fraction numerator cos space left parenthesis 2 x minus 2 right parenthesis minus 1 over denominator left parenthesis x minus 1 right parenthesis space tan space left parenthesis 3 x minus 3 right parenthesis end fraction equals...         

  1. negative 2 over 3      

  2. negative 1 third    

  3. 1      

  4. 1 1 half   

  5. 3    

Pembahasan Video:

Pembahasan:

Ingat kembali beberapa sifat berikut.

  •  a b plus a c equals a left parenthesis b plus c right parenthesis    

 

  • 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 end table  

 

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

 

  • limit as x rightwards arrow c of fraction numerator sin space a left parenthesis x minus 1 right parenthesis over denominator b left parenthesis x minus 1 right parenthesis end fraction equals limit as x rightwards arrow 0 of fraction numerator space a left parenthesis x minus 1 right parenthesis over denominator sin space b left parenthesis x minus 1 right parenthesis end fraction equals a over b

 

  • limit as x rightwards arrow c of fraction numerator sin space a left parenthesis x minus 1 right parenthesis over denominator tan space b left parenthesis x minus 1 right parenthesis end fraction equals a over b     

Dari aturan di atas, maka diperoleh

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 1 of fraction numerator cos space left parenthesis 2 x minus 2 right parenthesis minus 1 over denominator left parenthesis x minus 1 right parenthesis space tan space left parenthesis 3 x minus 3 right parenthesis end fraction end cell equals cell limit as x rightwards arrow 1 of fraction numerator cos space 2 left parenthesis x minus 1 right parenthesis minus 1 over denominator left parenthesis x minus 1 right parenthesis space tan space 3 left parenthesis x minus 1 right parenthesis end fraction end cell row blank equals cell limit as x rightwards arrow 1 of fraction numerator 1 minus 2 space sin squared space left parenthesis x minus 1 right parenthesis minus 1 over denominator left parenthesis x minus 1 right parenthesis space tan space 3 left parenthesis x minus 1 right parenthesis end fraction end cell row blank equals cell limit as x rightwards arrow 1 of fraction numerator negative 2 space sin squared space left parenthesis x minus 1 right parenthesis over denominator left parenthesis x minus 1 right parenthesis space tan space 3 left parenthesis x minus 1 right parenthesis end fraction end cell row blank equals cell limit as x rightwards arrow 1 of fraction numerator negative 2 space sin space left parenthesis x minus 1 right parenthesis space sin space left parenthesis x minus 1 right parenthesis over denominator left parenthesis x minus 1 right parenthesis space tan space 3 left parenthesis x minus 1 right parenthesis end fraction end cell row blank equals cell negative 2 limit as x rightwards arrow 1 of fraction numerator sin space left parenthesis x minus 1 right parenthesis over denominator left parenthesis x minus 1 right parenthesis end fraction times fraction numerator sin space left parenthesis x minus 1 right parenthesis over denominator tan space 3 left parenthesis x minus 1 right parenthesis end fraction end cell row blank equals cell negative 2 times limit as x rightwards arrow 1 of fraction numerator sin space left parenthesis x minus 1 right parenthesis over denominator left parenthesis x minus 1 right parenthesis end fraction limit as x rightwards arrow 1 of fraction numerator sin space left parenthesis x minus 1 right parenthesis over denominator tan space 3 left parenthesis x minus 1 right parenthesis end fraction end cell row blank equals cell negative 2 times 1 times 1 third end cell row blank equals cell negative 2 over 3 end cell end table end style   

 

Dengan demikian, limit as x rightwards arrow 1 of fraction numerator cos space left parenthesis 2 x minus 2 right parenthesis minus 1 over denominator left parenthesis x minus 1 right parenthesis space tan space left parenthesis 3 x minus 3 right parenthesis end fraction equals negative 2 over 3.

Jadi, jawaban yang benar adalah A.

 

Jawaban terverifikasi

Dijawab oleh:

N. Puspita

Terakhir diupdate 07 Oktober 2021

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