Tentukan nilai: x→2lim​[x2+2x−84x​−x−22​]

Pertanyaan

Tentukan nilai:

begin mathsize 14px style limit as straight x rightwards arrow 2 of open square brackets fraction numerator 4 straight x over denominator straight x squared plus 2 straight x minus 8 end fraction minus fraction numerator 2 over denominator straight x minus 2 end fraction close square brackets end style 

W. Lestari

Master Teacher

Mahasiswa/Alumni Universitas Sriwijaya

Jawaban terverifikasi

Jawaban

nilai begin mathsize 14px style limit as straight x rightwards arrow 2 of open square brackets fraction numerator 4 straight x over denominator straight x squared plus 2 straight x minus 8 end fraction minus fraction numerator 2 over denominator straight x minus 2 end fraction close square brackets end style adalah tidak ada.

Pembahasan

Perhatikan perhitungan berikut:

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as straight x rightwards arrow 2 of open square brackets fraction numerator 4 straight x over denominator straight x squared plus 2 straight x minus 8 end fraction minus fraction numerator 2 over denominator straight x minus 2 end fraction close square brackets end cell equals cell fraction numerator 4 times 2 over denominator 2 squared plus 2 times 2 minus 8 end fraction minus fraction numerator 2 over denominator 2 minus 2 end fraction end cell row blank equals cell fraction numerator 8 over denominator 4 plus 4 minus 8 end fraction minus 2 over 0 end cell row blank equals cell 8 over 0 minus 2 over 0 end cell row blank equals cell 6 over 0 end cell row blank equals infinity end table 

Karena hasil dari limit tersebut ketika disubstitusikan x equals 2 merupakan bilangan tak hingga 6 over 0, maka kita lakukan pemfaktoran pada bentuk limit tersebut sebagai berikut:

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as straight x rightwards arrow 2 of open square brackets fraction numerator 4 straight x over denominator straight x squared plus 2 straight x minus 8 end fraction minus fraction numerator 2 over denominator straight x minus 2 end fraction close square brackets end cell equals cell limit as straight x rightwards arrow 2 of open square brackets fraction numerator 4 straight x over denominator left parenthesis straight x plus 4 right parenthesis left parenthesis straight x minus 2 right parenthesis end fraction minus fraction numerator 2 over denominator straight x minus 2 end fraction close square brackets end cell row blank equals cell limit as straight x rightwards arrow 2 of open square brackets fraction numerator 4 straight x over denominator left parenthesis straight x plus 4 right parenthesis left parenthesis straight x minus 2 right parenthesis end fraction minus fraction numerator 2 left parenthesis straight x plus 4 right parenthesis over denominator left parenthesis straight x plus 4 right parenthesis left parenthesis straight x minus 2 right parenthesis end fraction close square brackets end cell row blank equals cell limit as straight x rightwards arrow 2 of open square brackets fraction numerator 4 straight x minus 2 left parenthesis straight x plus 4 right parenthesis over denominator left parenthesis straight x plus 4 right parenthesis left parenthesis straight x minus 2 right parenthesis end fraction close square brackets end cell row blank equals cell limit as straight x rightwards arrow 2 of open square brackets fraction numerator 4 straight x minus 2 straight x minus 8 over denominator left parenthesis straight x plus 4 right parenthesis left parenthesis straight x minus 2 right parenthesis end fraction close square brackets end cell row blank equals cell limit as straight x rightwards arrow 2 of open square brackets fraction numerator 2 straight x minus 8 over denominator left parenthesis straight x plus 4 right parenthesis left parenthesis straight x minus 2 right parenthesis end fraction close square brackets end cell row blank equals cell fraction numerator 2 times 2 minus 8 over denominator left parenthesis 2 plus 4 right parenthesis left parenthesis 2 minus 2 right parenthesis end fraction end cell row blank equals cell fraction numerator 4 minus 8 over denominator left parenthesis 6 right parenthesis left parenthesis 0 right parenthesis end fraction end cell row blank equals cell fraction numerator negative 4 over denominator 0 end fraction end cell row blank equals cell negative infinity end cell end table end style   

Karena hasil limit ketika di substitusikan x equals 2 merupakan bilangan tak hingga 6 over 0 dan ketika difaktorkan menghasilkan bilangan negatif tak hingga fraction numerator negative 4 over denominator 0 end fraction. Maka sekarang kita lakukan perhitungan limit dengan menenentukan nilai limit kiri dan limit kanannya sebagai berikut:

Dari tabel di atas, dapat dilihat bahwa nilai limit kiri dan limit kanan menuju ke arah bilangan yang berbeda sehingga nilai limit tersebut tidak ada.

Jadi, nilai begin mathsize 14px style limit as straight x rightwards arrow 2 of open square brackets fraction numerator 4 straight x over denominator straight x squared plus 2 straight x minus 8 end fraction minus fraction numerator 2 over denominator straight x minus 2 end fraction close square brackets end style adalah tidak ada.

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