Pertanyaan

Nilai dari x → ∞ lim 2 x tan ( x 1 ) . sec ( x 2 ) adalah ...

Nilai dari $x \to \infty lim 2 x tan (\frac{1}{x}) . sec (\frac{2}{x})$ adalah ...

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Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah C,

Pembahasan

Ingat bahwa : Misalkan , sehingga , maka . Perhatikan bahwa Jadi, jawaban yang tepat adalah C,

Ingat bahwa :

$begin mathsize 14px style limit as straight y rightwards arrow 0 of fraction numerator tan space straight y over denominator straight y end fraction equals 1 comma space sec space straight A equals fraction numerator 1 over denominator cos space straight A end fraction comma space straight a space. space straight b equals fraction numerator straight b over denominator begin display style 1 over straight a end style end fraction end style$

Misalkan undefined , sehingga , maka .

Perhatikan bahwa

$begin mathsize 14px style equals limit as straight x rightwards arrow infinity of 2 straight x space tan space open parentheses 1 over straight x close parentheses space. space sec space open parentheses 2 over straight x close parentheses equals limit as straight x rightwards arrow infinity of fraction numerator tan space open parentheses begin display style 1 over straight x end style close parentheses space 2 space sec space open parentheses begin display style 1 over straight x end style close parentheses over denominator begin display style 1 over straight x end style end fraction equals limit as straight x rightwards arrow infinity of fraction numerator tan space straight y space. space 2 space sec space 2 space straight y over denominator straight y end fraction equals limit as straight x rightwards arrow infinity of open parentheses fraction numerator tan space straight y over denominator straight y end fraction space. space 2 space sec space 2 space straight y close parentheses equals limit as straight x rightwards arrow infinity of open parentheses fraction numerator tan space straight y over denominator straight y end fraction space. space fraction numerator 2 over denominator cos space 2 straight y end fraction close parentheses equals 1 space. space fraction numerator 2 over denominator cos space 0 end fraction equals 1 space. space 2 over 1 equals 2 end style$

Jadi, jawaban yang tepat adalah C,