Hitunglah nilai dari: x→1lim​1−x1​[1−x1−x20​−20].

Pertanyaan

Hitunglah nilai dari:

limit as x rightwards arrow 1 of fraction numerator 1 over denominator 1 minus x end fraction open square brackets fraction numerator 1 minus x to the power of 20 over denominator 1 minus x end fraction minus 20 close square brackets.

H. Firmansyah

Master Teacher

Mahasiswa/Alumni Universitas Gadjah Mada

Jawaban terverifikasi

Jawaban

diperoleh limit as x rightwards arrow 1 of fraction numerator 1 over denominator 1 minus x end fraction open square brackets fraction numerator 1 minus x to the power of 20 over denominator 1 minus x end fraction minus 20 close square brackets equals negative 190.

Pembahasan

Langkah pertama untuk menyelesaikan limit diatas dengan mensubstitusikan x equals 1 ke bentuk aljabarnya, diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 1 of fraction numerator 1 over denominator 1 minus x end fraction open square brackets fraction numerator 1 minus x to the power of 20 over denominator 1 minus x end fraction minus 20 close square brackets end cell equals cell fraction numerator 1 over denominator 1 minus 1 end fraction open square brackets fraction numerator 1 minus 1 to the power of 20 over denominator 1 minus 1 end fraction minus 20 close square brackets end cell row blank equals cell 1 over 0 left parenthesis negative 20 right parenthesis end cell row blank equals cell negative 20 over 0 end cell row blank equals cell negative infinity end cell end table 

Karena merupakan salah satu bentuk tentu, maka dapat dicari nilainya dengan cara lain

Perhatikan

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as x rightwards arrow 1 of fraction numerator 1 over denominator 1 minus x end fraction open square brackets fraction numerator 1 minus x to the power of 20 over denominator 1 minus x end fraction minus 20 close square brackets divided by end cell row blank equals cell limit as x rightwards arrow 1 of fraction numerator up diagonal strike open parentheses x minus 1 close parentheses end strike open parentheses x to the power of 18 plus 2 x to the power of 17 plus 3 x to the power of 16 plus 4 x to the power of 15 plus 5 x to the power of 14 plus 6 x to the power of 13 plus 7 x to the power of 12 plus 8 x to the power of 11 plus 9 x to the power of 10 plus 10 x to the power of 9 plus 11 x to the power of 8 plus 12 x to the power of 7 plus 13 x to the power of 6 plus 14 x to the power of 5 plus 15 x to the power of 4 plus 16 x cubed plus 17 x squared plus 18 x plus 19 close parentheses over denominator open parentheses negative 1 close parentheses up diagonal strike open parentheses x minus 1 close parentheses end strike end fraction end cell row blank equals cell negative open parentheses 1 to the power of 18 plus 2 left parenthesis 1 right parenthesis to the power of 17 plus 3 left parenthesis 1 right parenthesis to the power of 16 plus 4 left parenthesis 1 right parenthesis to the power of 15 plus 5 left parenthesis 1 right parenthesis to the power of 14 plus 6 left parenthesis 1 right parenthesis to the power of 13 plus 7 left parenthesis 1 right parenthesis to the power of 12 plus 8 left parenthesis 1 right parenthesis to the power of 11 plus 9 left parenthesis 1 right parenthesis to the power of 10 plus 10 left parenthesis 1 right parenthesis to the power of 9 plus 11 left parenthesis 1 right parenthesis to the power of 8 plus 12 left parenthesis 1 right parenthesis to the power of 7 plus 13 left parenthesis 1 right parenthesis to the power of 6 plus 14 left parenthesis 1 right parenthesis to the power of 5 plus 15 left parenthesis 1 right parenthesis to the power of 4 plus 16 left parenthesis 1 right parenthesis cubed plus 17 left parenthesis 1 right parenthesis squared plus 18 left parenthesis 1 right parenthesis plus 19 close parentheses end cell row blank equals cell negative 190 end cell end table

Dengan demikian, diperoleh limit as x rightwards arrow 1 of fraction numerator 1 over denominator 1 minus x end fraction open square brackets fraction numerator 1 minus x to the power of 20 over denominator 1 minus x end fraction minus 20 close square brackets equals negative 190.

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