Hitunglah nilai dari: x→1lim1−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 $x=1$ 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 $limit as x rightwards arrow 1 of fraction numerator 1 over denominator 1 minus x end fraction open square brackets fraction numerator 1 minus x to the power of 20 over denominator 1 minus x end fraction minus 20 close square brackets equals negative 190$ .