Pertanyaan

Nilai dari adalah ....

Nilai dari $begin mathsize 14px style limit as x rightwards arrow straight infinity of invisible function application fraction numerator open parentheses 1 minus csc squared invisible function application fraction numerator 2 over denominator square root of x end fraction close parentheses over denominator x end fraction end style$ adalah ....

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Klaim

H. Nufus

Master Teacher

Mahasiswa/Alumni Universitas Negeri Surabaya

Jawaban terverifikasi

Pembahasan

Misalkan , maka . Sehingga ketika , maka . Maka

Misalkan $begin mathsize 14px style y equals fraction numerator 1 over denominator square root of x end fraction end style$ , maka .

Sehingga ketika , maka .

Maka

$begin mathsize 14px style limit as x rightwards arrow straight infinity of invisible function application fraction numerator open parentheses 1 minus csc squared invisible function application fraction numerator 2 over denominator square root of x end fraction close parentheses over denominator x end fraction equals limit as x rightwards arrow straight infinity of invisible function application fraction numerator open parentheses 1 minus csc squared invisible function application 2 open parentheses fraction numerator 1 over denominator square root of x end fraction close parentheses close parentheses over denominator x end fraction equals limit as y rightwards arrow 0 of invisible function application fraction numerator open parentheses 1 minus csc squared invisible function application 2 y close parentheses over denominator 1 over y squared end fraction equals limit as y rightwards arrow 0 of invisible function application y squared open parentheses 1 minus csc squared invisible function application 2 y close parentheses equals limit as y rightwards arrow 0 of invisible function application y squared open parentheses negative cot squared invisible function application 2 y close parentheses equals limit as y rightwards arrow 0 of invisible function application open parentheses negative y squared cot squared invisible function application 2 y close parentheses equals limit as y rightwards arrow 0 of invisible function application open parentheses negative y squared times fraction numerator 1 over denominator tan squared invisible function application 2 y end fraction close parentheses equals limit as y rightwards arrow 0 of invisible function application open parentheses negative fraction numerator y squared over denominator tan squared invisible function application 2 y end fraction close parentheses equals limit as y rightwards arrow 0 of invisible function application open parentheses negative fraction numerator y over denominator tan invisible function application 2 y end fraction times fraction numerator y over denominator tan invisible function application 2 y end fraction close parentheses equals limit as y rightwards arrow 0 of invisible function application open parentheses negative fraction numerator y over denominator tan invisible function application 2 y end fraction times fraction numerator y over denominator tan invisible function application 2 y end fraction times 2 over 2 times 2 over 2 close parentheses equals limit as y rightwards arrow 0 of invisible function application open parentheses negative fraction numerator 2 y over denominator tan invisible function application 2 y end fraction times fraction numerator 2 y over denominator tan invisible function application 2 y end fraction times 1 half times 1 half close parentheses equals negative 1 times 1 times 1 half times 1 half equals negative 1 fourth end style$