Pertanyaan

Nilai dari adalah ...

Nilai dari $begin mathsize 14px style limit as x rightwards arrow straight infinity of invisible function application fraction numerator square root of 5 minus square root of fraction numerator 2 plus 5 x over denominator x end fraction end root over denominator sin invisible function application open parentheses 3 over x close parentheses end fraction end style$ adalah ...

8 dari 10 siswa nilainya naik

dengan paket belajar pilihan

Habis dalam

Klaim

H. Nufus

Master Teacher

Mahasiswa/Alumni Universitas Negeri Surabaya

Jawaban terverifikasi

Pembahasan

Perhatikan bahwa Misalkan . Ketika x → ∞ , maka y → 0 . Sehingga

Perhatikan bahwa

$begin mathsize 14px style limit as x rightwards arrow straight infinity of invisible function application fraction numerator square root of 5 minus square root of fraction numerator 2 plus 5 x over denominator x end fraction end root over denominator sin invisible function application open parentheses 3 over x close parentheses end fraction equals limit as x rightwards arrow straight infinity of invisible function application fraction numerator square root of 5 minus square root of 2 over x plus 5 end root over denominator sin invisible function application open parentheses 3 over x close parentheses end fraction equals limit as x rightwards arrow straight infinity of invisible function application open parentheses fraction numerator square root of 5 minus square root of 2 over x plus 5 end root over denominator sin invisible function application open parentheses 3 over x close parentheses end fraction times fraction numerator square root of 5 plus square root of 2 over x plus 5 end root over denominator square root of 5 plus square root of 2 over x plus 5 end root end fraction close parentheses equals limit as x rightwards arrow straight infinity of invisible function application open parentheses fraction numerator 5 minus open parentheses 2 over x plus 5 close parentheses over denominator sin invisible function application open parentheses 3 over x close parentheses end fraction times fraction numerator 1 over denominator square root of 5 plus square root of 2 over x plus 5 end root end fraction close parentheses equals limit as x rightwards arrow straight infinity of invisible function application open parentheses fraction numerator negative 2 over x over denominator sin invisible function application open parentheses 3 over x close parentheses end fraction times fraction numerator 1 over denominator square root of 5 plus square root of 2 over x plus 5 end root end fraction close parentheses equals limit as x rightwards arrow straight infinity of invisible function application fraction numerator negative 2 over x over denominator sin invisible function application open parentheses 3 over x close parentheses end fraction times limit as x rightwards arrow straight infinity of invisible function application fraction numerator 1 over denominator square root of 5 plus square root of 2 over x plus 5 end root end fraction equals limit as x rightwards arrow straight infinity of invisible function application open parentheses fraction numerator negative 2 over x over denominator sin invisible function application open parentheses 3 over x close parentheses end fraction times 3 over 3 close parentheses times fraction numerator 1 over denominator square root of 5 plus square root of 0 plus 5 end root end fraction equals limit as x rightwards arrow straight infinity of invisible function application open parentheses fraction numerator 3 over x over denominator sin invisible function application open parentheses 3 over x close parentheses end fraction times fraction numerator negative 2 over denominator 3 end fraction close parentheses times fraction numerator 1 over denominator square root of 5 plus square root of 5 end fraction equals limit as x rightwards arrow straight infinity of invisible function application open parentheses fraction numerator 3 over x over denominator sin invisible function application open parentheses 3 over x close parentheses end fraction times fraction numerator negative 2 over denominator 3 end fraction close parentheses times fraction numerator 1 over denominator 2 square root of 5 end fraction end style$

Misalkan . Ketika x → ∞, maka y → 0. Sehingga

$begin mathsize 14px style limit as x rightwards arrow straight infinity of invisible function application open parentheses fraction numerator 3 over x over denominator sin invisible function application open parentheses 3 over x close parentheses end fraction times fraction numerator negative 2 over denominator 3 end fraction close parentheses times fraction numerator 1 over denominator 2 square root of 5 end fraction equals limit as x rightwards arrow straight infinity of invisible function application open parentheses fraction numerator 3 open parentheses 1 over x close parentheses over denominator sin invisible function application 3 open parentheses 1 over x close parentheses end fraction times fraction numerator negative 2 over denominator 3 end fraction close parentheses times fraction numerator 1 over denominator 2 square root of 5 end fraction equals limit as y rightwards arrow 0 of invisible function application open parentheses fraction numerator 3 y over denominator sin invisible function application 3 y end fraction times fraction numerator negative 2 over denominator 3 end fraction close parentheses times fraction numerator 1 over denominator 2 square root of 5 end fraction equals 1 times fraction numerator negative 2 over denominator 3 end fraction times fraction numerator 1 over denominator 2 square root of 5 end fraction equals negative fraction numerator 1 over denominator 3 square root of 5 end fraction times fraction numerator square root of 5 over denominator square root of 5 end fraction equals negative 1 over 15 square root of 5 end style$