Pertanyaan

Nilai dari: x → 0 lim sin 3 x − 3 sin x sin 4 x − sin 3 x − sin x = ....

Nilai dari: $x \to 0 lim \frac{sin 4 x - sin 3 x - sin x}{sin 3 x - 3 sin x} = ....$

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Jawaban terverifikasi

Jawaban

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as straight x rightwards arrow 0 of end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator sin 4 straight x minus sin 3 straight x minus sinx over denominator sin space 3 straight x minus 3 sin space straight x end fraction end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 3 over 2 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank. end table end style$

Pembahasan

Menentukan nilai limit trigonometeri. Jadi,

Menentukan nilai limit trigonometeri.

$table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as straight x rightwards arrow 0 of fraction numerator sin 4 straight x minus sin 3 straight x minus sinx over denominator sin space 3 straight x minus 3 sin space straight x end fraction end cell row blank equals cell limit as straight x rightwards arrow 0 of fraction numerator sin space 4 straight x minus left parenthesis sin space 3 straight x plus sin space straight x right parenthesis over denominator sin space 3 straight x minus sin space straight x minus 2 sin space straight x end fraction end cell row blank equals cell limit as straight x rightwards arrow 0 of fraction numerator sin left parenthesis 4 x right parenthesis minus 2 sin begin display style fraction numerator 4 straight x over denominator 2 end fraction end style cos begin display style fraction numerator 2 straight x over denominator 2 end fraction end style over denominator 2 cos left parenthesis begin display style fraction numerator 4 straight x over denominator 2 end fraction end style right parenthesis sin left parenthesis begin display style fraction numerator 2 straight x over denominator 2 end fraction end style right parenthesis minus 2 sin space straight x end fraction end cell row blank equals cell limit as straight x rightwards arrow 0 of fraction numerator sin left parenthesis 4 x right parenthesis minus 2 sin begin display style 2 straight x times end style cos begin display style straight x end style over denominator 2 cos left parenthesis begin display style 2 straight x end style right parenthesis sin left parenthesis straight x right parenthesis minus 2 sin space straight x end fraction end cell row blank equals cell limit as straight x rightwards arrow 0 of fraction numerator 2 sin 2 straight x times cos 2 straight x minus 2 sin begin display style 2 straight x times end style cos begin display style straight x end style over denominator 2 cos left parenthesis begin display style 2 straight x end style right parenthesis times sin left parenthesis straight x right parenthesis minus 2 sin space straight x end fraction end cell row blank equals cell limit as straight x rightwards arrow 0 of fraction numerator 2 sin 2 straight x left parenthesis cos 2 straight x minus cos begin display style straight x right parenthesis end style over denominator 2 sin space straight x left parenthesis cos begin display style 2 straight x end style minus 1 right parenthesis end fraction end cell row blank equals cell limit as straight x rightwards arrow 0 of fraction numerator 2 sin 2 straight x left parenthesis negative 2 sin begin display style fraction numerator 3 straight x over denominator 2 end fraction end style sin begin display style straight x over 2 end style right parenthesis over denominator 2 sin space straight x times left parenthesis cos left parenthesis begin display style 2 end style begin display style x end style right parenthesis minus 1 right parenthesis end fraction end cell row blank equals cell limit as straight x rightwards arrow 0 of fraction numerator negative 4 sin 2 straight x times sin begin display style fraction numerator 3 straight x over denominator 2 end fraction times end style sin begin display style straight x over 2 end style over denominator 2 sinx times left parenthesis 1 minus 2 sin squared straight x minus 1 right parenthesis end fraction end cell row blank equals cell limit as straight x rightwards arrow 0 of fraction numerator negative 4 sin 2 straight x times sin begin display style fraction numerator 3 straight x over denominator 2 end fraction times end style sin begin display style straight x over 2 end style over denominator 2 sinx times left parenthesis negative 2 sin squared straight x right parenthesis end fraction end cell row blank equals cell limit as straight x rightwards arrow 0 of fraction numerator negative 4 sin 2 straight x times sin begin display style fraction numerator 3 straight x over denominator 2 end fraction times end style sin begin display style straight x over 2 end style over denominator negative 4 sinx times sinx times sinx end fraction end cell row blank equals cell space limit as straight x rightwards arrow 0 of fraction numerator sin 2 straight x times sin begin display style fraction numerator 3 straight x over denominator 2 end fraction times end style sin begin display style straight x over 2 end style over denominator sinx times sinx times sinx end fraction end cell row blank equals cell limit as straight x rightwards arrow 0 of fraction numerator sin 2 straight x over denominator sinx end fraction times limit as straight x rightwards arrow 0 of fraction numerator sin begin display style fraction numerator 3 straight x over denominator 2 end fraction end style over denominator sin space straight x end fraction times limit as straight x rightwards arrow 0 of fraction numerator sin begin display style straight x over 2 end style over denominator sinx end fraction end cell row blank equals cell 2 times 3 over 2 times 1 half end cell row blank equals cell 3 over 2 end cell end table$

Jadi, $begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as straight x rightwards arrow 0 of end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator sin 4 straight x minus sin 3 straight x minus sinx over denominator sin space 3 straight x minus 3 sin space straight x end fraction end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 3 over 2 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank. end table end style$