Pertanyaan

x → 0 lim 3 1 + x − 1 1 + x − 1

$x \to 0 lim \frac{1 + x - 1}{3 1 + x - 1}$

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Habis dalam

Klaim

I. Kumaralalita

Master Teacher

Mahasiswa/Alumni Universitas Gadjah Mada

Jawaban terverifikasi

Jawaban

nilai dari adalah .

nilai dari $begin mathsize 14px style limit as x rightwards arrow 0 of fraction numerator square root of 1 plus x end root minus 1 over denominator cube root of 1 plus x end root minus 1 end fraction end style$ adalah

Pembahasan

Nilai limit dapat dihitung dengan mensubstitusikan ke dalam fungsi seperti berikut, Nilai limit yang didapatkan adalah yang artinya tidak terdefinsi. Agar limit tersebut terdefinisi, maka terlebih dahulu dilakukan langkah L'hospital dengan menurunkan masing-masing pembilang dan penyebut. Turunan dari pembilang yaitu, Turunkan dari penyebut yaitu, Nilai limit adalah Jadi, nilai dari adalah .

Nilai limit dapat dihitung dengan mensubstitusikan ke dalam fungsi seperti berikut,

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 0 of space fraction numerator square root of 1 plus x end root minus 1 over denominator cube root of 1 plus x end root minus 1 end fraction end cell equals cell fraction numerator square root of 1 plus 0 end root minus 1 over denominator cube root of 1 plus 0 end root minus 1 end fraction end cell row blank equals cell fraction numerator 1 minus 1 over denominator 1 minus 1 end fraction end cell row blank equals cell 0 over 0 end cell end table end style$

Nilai limit yang didapatkan adalah yang artinya tidak terdefinsi. Agar limit tersebut terdefinisi, maka terlebih dahulu dilakukan langkah L'hospital dengan menurunkan masing-masing pembilang dan penyebut.

Turunan dari pembilang yaitu,

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator d open parentheses square root of 1 plus x end root minus 1 close parentheses over denominator d x end fraction end cell equals cell fraction numerator d open parentheses left parenthesis 1 plus x right parenthesis to the power of begin display style 1 half end style end exponent minus 1 close parentheses over denominator d x end fraction end cell row blank equals cell 1 half times left parenthesis 1 plus x right parenthesis to the power of 1 half minus 1 end exponent times 1 end cell row blank equals cell 1 half times left parenthesis 1 plus x right parenthesis to the power of negative 1 half end exponent end cell row blank equals cell fraction numerator 1 over denominator 2 square root of 1 plus x end root end fraction end cell end table end style$

Turunkan dari penyebut yaitu,

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator d open parentheses cube root of 1 plus x end root minus 1 close parentheses over denominator d x end fraction end cell equals cell fraction numerator d open parentheses left parenthesis 1 plus x right parenthesis to the power of begin display style 1 third end style end exponent minus 1 close parentheses over denominator d x end fraction end cell row blank equals cell 1 third times left parenthesis 1 plus x right parenthesis to the power of 1 third minus 1 end exponent times 1 end cell row blank equals cell 1 third times left parenthesis 1 plus x right parenthesis blank to the power of negative 2 over 3 end exponent times 1 end cell row blank equals cell fraction numerator 1 over denominator 3 space cube root of open parentheses 1 plus x close parentheses squared end root end fraction end cell end table end style$

Nilai limit adalah

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 0 of fraction numerator square root of 1 plus x end root minus 1 over denominator cube root of 1 plus x end root minus 1 end fraction end cell equals cell limit as x rightwards arrow 0 of fraction numerator begin display style fraction numerator d open parentheses square root of 1 plus x end root minus 1 close parentheses over denominator d x end fraction end style over denominator begin display style fraction numerator d open parentheses cube root of 1 plus x end root minus 1 close parentheses over denominator d x end fraction end style end fraction end cell row blank equals cell limit as x rightwards arrow 0 of fraction numerator begin display style fraction numerator 1 over denominator 2 square root of 1 plus x end root end fraction end style over denominator begin display style fraction numerator 1 over denominator 3 space cube root of left parenthesis 1 plus x right parenthesis squared end root end fraction end style end fraction end cell row blank equals cell limit as x rightwards arrow 0 of space fraction numerator 1 over denominator 2 square root of 1 plus x end root end fraction times fraction numerator 3 space cube root of left parenthesis 1 plus x right parenthesis squared end root over denominator 1 end fraction end cell row blank equals cell limit as x rightwards arrow 0 of fraction numerator 3 space cube root of left parenthesis 1 plus x right parenthesis squared end root over denominator 2 square root of 1 plus x end root end fraction end cell row blank equals cell fraction numerator 3 space cube root of left parenthesis 1 plus 0 right parenthesis squared end root over denominator 2 square root of 1 plus 0 end root end fraction end cell row blank equals cell fraction numerator 3 times cube root of 1 over denominator 2 times square root of 1 end fraction end cell row blank equals cell 3 over 2 end cell end table end style$

Jadi, nilai dari $begin mathsize 14px style limit as x rightwards arrow 0 of fraction numerator square root of 1 plus x end root minus 1 over denominator cube root of 1 plus x end root minus 1 end fraction end style$ adalah .