Pertanyaan

Selesaikan setiap bentuk limit berikut ini. d. x → 1 lim [ x 2 − 1 2 − x − 1 2 ]

Selesaikan setiap bentuk limit berikut ini.

d. $x \to 1 lim [\frac{2}{x ^{2} - 1} - \frac{2}{x - 1}]$

E. Dwi

Master Teacher

Mahasiswa/Alumni Universitas Sriwijaya

Jawaban terverifikasi

Pembahasan

Limit merupakan nilai yang sangan dekat dengan suatu variabel tertentu. Baik pendekatannya dilakukan dari sbelah kiri variabel atau sebelah kanan variabel . Dengan menggunakan pendekatan dari kiridan kanan, maka nilai dapat ditentukan seperti berikut: Limit kiri: Limit kanan: Jadi, hasil limit dari dengan menggunakan limit kiri dan limit kanan secara berturut-turut adalah dan .

Limit merupakan nilai yang sangan dekat dengan suatu variabel tertentu. Baik pendekatannya dilakukan dari sbelah kiri variabel atau sebelah kanan variabel .

Dengan menggunakan pendekatan dari kiri dan kanan, maka nilai $begin mathsize 14px style limit as x rightwards arrow 1 of open square brackets fraction numerator 2 over denominator x squared minus 1 end fraction minus fraction numerator 2 over denominator x minus 1 end fraction close square brackets end style$ dapat ditentukan seperti berikut:

Limit kiri:

$table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 1 to the power of minus of open square brackets fraction numerator 2 over denominator x squared minus 1 end fraction minus fraction numerator 2 over denominator x minus 1 end fraction close square brackets end cell equals cell limit as x rightwards arrow 1 to the power of minus of open square brackets fraction numerator 2 minus 2 open parentheses x plus 1 close parentheses over denominator left parenthesis x minus 1 right parenthesis left parenthesis x plus 1 right parenthesis end fraction close square brackets end cell row blank equals cell limit as x rightwards arrow 1 to the power of minus of open square brackets negative fraction numerator 2 x over denominator left parenthesis x minus 1 right parenthesis left parenthesis x plus 1 right parenthesis end fraction close square brackets end cell row blank equals cell negative 2 limit as x rightwards arrow 1 to the power of minus of open square brackets fraction numerator x over denominator left parenthesis x minus 1 right parenthesis left parenthesis x plus 1 right parenthesis end fraction close square brackets end cell row blank equals cell negative 2 limit as x rightwards arrow 1 to the power of minus of open square brackets fraction numerator 1 over denominator left parenthesis x minus 1 right parenthesis end fraction close square brackets limit as x rightwards arrow 1 to the power of minus of open square brackets fraction numerator x over denominator left parenthesis x plus 1 right parenthesis end fraction close square brackets end cell row blank equals cell negative 2 limit as x rightwards arrow 1 to the power of minus of open square brackets fraction numerator 1 over denominator left parenthesis x minus 1 right parenthesis end fraction close square brackets open square brackets fraction numerator 1 over denominator 1 plus 1 end fraction close square brackets end cell row blank equals cell negative 2 limit as x rightwards arrow 1 to the power of minus of open square brackets fraction numerator 1 over denominator left parenthesis x minus 1 right parenthesis end fraction close square brackets 1 half end cell row blank equals cell negative 1 limit as x rightwards arrow 1 to the power of minus of open square brackets fraction numerator 1 over denominator left parenthesis x minus 1 right parenthesis end fraction close square brackets end cell row blank equals cell negative 1 open square brackets negative infinity close square brackets end cell row blank equals infinity end table$

Limit kanan:

$table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 1 to the power of plus of open square brackets fraction numerator 2 over denominator x squared minus 1 end fraction minus fraction numerator 2 over denominator x minus 1 end fraction close square brackets end cell equals cell limit as x rightwards arrow 1 to the power of plus of open square brackets fraction numerator 2 minus 2 open parentheses x plus 1 close parentheses over denominator left parenthesis x minus 1 right parenthesis left parenthesis x plus 1 right parenthesis end fraction close square brackets end cell row blank equals cell limit as x rightwards arrow 1 plus of open square brackets negative fraction numerator 2 x over denominator left parenthesis x minus 1 right parenthesis left parenthesis x plus 1 right parenthesis end fraction close square brackets end cell row blank equals cell negative 2 limit as x rightwards arrow 1 to the power of plus of open square brackets fraction numerator x over denominator left parenthesis x minus 1 right parenthesis left parenthesis x plus 1 right parenthesis end fraction close square brackets end cell row blank equals cell negative 2 limit as x rightwards arrow 1 to the power of plus of open square brackets fraction numerator 1 over denominator left parenthesis x minus 1 right parenthesis end fraction close square brackets limit as x rightwards arrow 1 to the power of plus of open square brackets fraction numerator x over denominator left parenthesis x plus 1 right parenthesis end fraction close square brackets end cell row blank equals cell negative 2 limit as x rightwards arrow 1 to the power of plus of open square brackets fraction numerator 1 over denominator left parenthesis x minus 1 right parenthesis end fraction close square brackets open square brackets fraction numerator 1 over denominator 1 plus 1 end fraction close square brackets end cell row blank equals cell negative 2 limit as x rightwards arrow 1 to the power of plus of open square brackets fraction numerator 1 over denominator left parenthesis x minus 1 right parenthesis end fraction close square brackets 1 half end cell row blank equals cell negative 1 limit as x rightwards arrow 1 to the power of plus of open square brackets fraction numerator 1 over denominator left parenthesis x minus 1 right parenthesis end fraction close square brackets end cell row blank equals cell negative 1 open square brackets infinity close square brackets end cell row blank equals cell negative infinity end cell end table$

Jadi, hasil limit dari $begin mathsize 14px style limit as x rightwards arrow 1 of open square brackets fraction numerator 2 over denominator x squared minus 1 end fraction minus fraction numerator 2 over denominator x minus 1 end fraction close square brackets end style$ dengan menggunakan limit kiri dan limit kanan secara berturut-turut adalah infinity dan negative infinity .