Pertanyaan

Tentukan nilai limit berikut dengan dalil L'Hospital. x → 1 lim 3 x + 7 − 2 x − 1

Tentukan nilai limit berikut dengan dalil L'Hospital.

$x \to 1 lim \frac{x - 1}{3 x + 7 - 2}$

W. Lestari

Master Teacher

Mahasiswa/Alumni Universitas Sriwijaya

Jawaban terverifikasi

Jawaban

.

$limit as x rightwards arrow 1 of fraction numerator square root of x minus 1 over denominator cube root of x plus 7 end root minus 2 end fraction equals 6$ .

Pembahasan

Nilai limit tersebut dengan dalil L'Hospital dapat ditentukan sebagai berikut: Jadi, .

Nilai limit tersebut dengan dalil L'Hospital dapat ditentukan sebagai berikut:

$table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 1 of fraction numerator square root of x minus 1 over denominator cube root of x plus 7 end root minus 2 end fraction end cell equals cell limit as x rightwards arrow 1 of fraction numerator x to the power of begin display style 1 half end style end exponent minus 1 over denominator open parentheses x plus 7 close parentheses to the power of begin display style 1 third end style end exponent minus 2 end fraction end cell row blank equals cell limit as x rightwards arrow 1 of fraction numerator begin display style 1 half end style times x to the power of begin display style 1 half minus 1 end style end exponent minus 0 over denominator begin display style 1 third end style times open parentheses x plus 7 close parentheses to the power of begin display style 1 third minus 1 end style end exponent times 1 minus 0 end fraction end cell row blank equals cell limit as x rightwards arrow 1 of fraction numerator begin display style 1 half end style x to the power of begin display style negative 1 half end style end exponent over denominator begin display style 1 third end style open parentheses x plus 7 close parentheses to the power of begin display style negative 2 over 3 end style end exponent end fraction end cell row blank equals cell fraction numerator begin display style 1 half times open parentheses 1 close parentheses to the power of negative 1 half end exponent end style over denominator begin display style 1 third times open parentheses 1 plus 7 close parentheses to the power of negative 2 over 3 end exponent end style end fraction end cell row blank equals cell fraction numerator begin display style 1 half times 1 over 1 to the power of begin display style 1 half end style end exponent end style over denominator begin display style 1 third times open parentheses 8 close parentheses to the power of negative 2 over 3 end exponent end style end fraction end cell row blank equals cell fraction numerator begin display style 1 half times fraction numerator 1 over denominator square root of 1 end fraction end style over denominator begin display style 1 third times open parentheses 2 to the power of up diagonal strike 3 end exponent close parentheses to the power of negative fraction numerator 2 over denominator up diagonal strike 3 end fraction end exponent end style end fraction end cell row blank equals cell fraction numerator begin display style 1 half times 1 end style over denominator begin display style 1 third times 2 to the power of negative 2 end exponent end style end fraction end cell row blank equals cell fraction numerator begin display style 1 half end style over denominator begin display style 1 third end style times begin display style 1 over 2 squared end style end fraction end cell row blank equals cell fraction numerator begin display style 1 half end style over denominator begin display style 1 third end style times begin display style 1 fourth end style end fraction end cell row blank equals cell fraction numerator begin display style 1 half end style over denominator begin display style 1 over 12 end style end fraction end cell row blank equals cell 1 half times 12 over 1 end cell row blank equals 6 end table$

Jadi, $limit as x rightwards arrow 1 of fraction numerator square root of x minus 1 over denominator cube root of x plus 7 end root minus 2 end fraction equals 6$ .