Pertanyaan

Nilai dari adalah ....

Nilai dari $begin mathsize 14px style limit as x rightwards arrow straight infinity of invisible function application fraction numerator 1 minus cos invisible function application open parentheses 4 over x close parentheses over denominator x to the power of negative 2 end exponent end fraction end style$ adalah ....

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Perhatikan bahwa Sehingga Misalkan , maka . Ketika , maka . Sehingga

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Sehingga

$begin mathsize 14px style limit as x rightwards arrow straight infinity of invisible function application fraction numerator 1 minus cos invisible function application open parentheses 4 over x close parentheses over denominator x to the power of negative 2 end exponent end fraction equals limit as x rightwards arrow straight infinity of invisible function application open parentheses open parentheses 1 minus cos invisible function application open parentheses 4 over x close parentheses close parentheses times x squared close parentheses equals limit as x rightwards arrow straight infinity of invisible function application open parentheses open parentheses 1 minus open parentheses 1 minus 2 sin squared invisible function application 1 half open parentheses 4 over x close parentheses close parentheses close parentheses times x squared close parentheses equals limit as x rightwards arrow straight infinity of invisible function application open parentheses open parentheses 2 sin squared invisible function application open parentheses 2 over x close parentheses close parentheses times x squared close parentheses equals limit as x rightwards arrow straight infinity of invisible function application open parentheses open parentheses 2 sin squared invisible function application 2 open parentheses 1 over x close parentheses close parentheses times x squared close parentheses end style$

Misalkan , maka . Ketika , maka .

Sehingga

$begin mathsize 14px style limit as x rightwards arrow straight infinity of invisible function application open parentheses open parentheses 2 sin squared invisible function application 2 open parentheses 1 over x close parentheses close parentheses times x squared close parentheses equals limit as y rightwards arrow 0 of invisible function application open parentheses open parentheses 2 sin squared invisible function application 2 y close parentheses times open parentheses 1 over y close parentheses squared close parentheses equals limit as y rightwards arrow 0 of invisible function application open parentheses open parentheses 2 sin squared invisible function application 2 y close parentheses times 1 over y squared close parentheses equals limit as y rightwards arrow 0 of invisible function application open parentheses fraction numerator 2 sin squared invisible function application 2 y over denominator y squared end fraction close parentheses equals limit as y rightwards arrow 0 of invisible function application open parentheses fraction numerator 2 sin invisible function application 2 y times sin invisible function application 2 y over denominator y times y end fraction close parentheses equals limit as y rightwards arrow 0 of invisible function application open parentheses fraction numerator 2 sin invisible function application 2 y over denominator y end fraction times fraction numerator sin invisible function application 2 y over denominator y end fraction close parentheses equals limit as y rightwards arrow 0 of invisible function application open parentheses fraction numerator 2 sin invisible function application 2 y over denominator y end fraction times 2 over 2 times fraction numerator sin invisible function application 2 y over denominator y end fraction times 2 over 2 close parentheses equals limit as y rightwards arrow 0 of invisible function application open parentheses 2 times fraction numerator sin invisible function application 2 y over denominator 2 y end fraction times 2 times fraction numerator sin invisible function application 2 y over denominator 2 y end fraction times 2 close parentheses equals 2 times 1 times 2 times 1 times 2 equals 8 end style$