Pertanyaan

x → ∞ lim 1 − cos ( x 1 ) sin 2 ( x 1 )

$x \to \infty lim \frac{sin ^{2} ( \frac{1}{x} )}{1 - cos ( \frac{1}{x} )}$

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Jawaban terverifikasi

Jawaban

$begin mathsize 14px style limit as x rightwards arrow infinity of fraction numerator sin squared begin display style left parenthesis 1 over x right parenthesis end style over denominator 1 minus cos begin display style left parenthesis 1 over x right parenthesis end style end fraction equals 2 end style$

Pembahasan

Misal: Limit fungsi trigonometri tersebut dapat ditulis sebagai berikut. Dengan demikian,

$begin mathsize 14px style limit as x rightwards arrow infinity of fraction numerator sin squared open parentheses begin display style 1 over x end style close parentheses over denominator 1 minus cos open parentheses begin display style 1 over x end style close parentheses end fraction end style$

Misal:

Limit fungsi trigonometri tersebut dapat ditulis sebagai berikut.

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as y rightwards arrow 0 of fraction numerator sin squared y over denominator 1 minus cos space y end fraction end cell equals cell limit as y rightwards arrow 0 of fraction numerator sin squared y over denominator 1 minus open parentheses 1 minus 2 sin squared begin display style 1 half end style y close parentheses end fraction end cell row blank equals cell limit as y rightwards arrow 0 of fraction numerator sin squared y over denominator 2 sin squared 1 half y end fraction end cell row blank equals cell 1 half times limit as y rightwards arrow 0 of fraction numerator sin squared y over denominator y squared end fraction times y squared times limit as y rightwards arrow 0 of fraction numerator open parentheses begin display style 1 half end style y close parentheses squared over denominator sin squared 1 half y end fraction times 1 over open parentheses 1 half y close parentheses squared end cell row blank equals cell 1 half times 1 times y squared times 1 times 4 over y squared end cell row blank equals 2 end table end style$

Dengan demikian, $begin mathsize 14px style limit as x rightwards arrow infinity of fraction numerator sin squared begin display style left parenthesis 1 over x right parenthesis end style over denominator 1 minus cos begin display style left parenthesis 1 over x right parenthesis end style end fraction equals 2 end style$