Pertanyaan

x → 0 lim 3 x 2 sin 2 x cos x 7 tan 4 x = ...

$x \to 0 lim \frac{7 tan ^{4} x}{3 x ^{2} sin ^{2} x cos x} = ...$

N. Puspita

Master Teacher

Jawaban terverifikasi

Jawaban

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$table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as x rightwards arrow 0 of end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 7 space tan to the power of 4 space x over denominator 3 x squared space sin squared space x space cos space x end fraction end cell end table equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 7 over 3 end cell end table$ .

Pembahasan

Ingat! Karena , maka nilai dari adalah: Jadi, .

Ingat!

$table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 0 of fraction numerator x over denominator sin space x end fraction end cell equals cell limit as x rightwards arrow 0 of fraction numerator sin space x over denominator x end fraction equals 1 end cell row cell limit as x rightwards arrow 0 of fraction numerator x over denominator tan space x end fraction end cell equals cell limit as x rightwards arrow 0 of fraction numerator tan space x over denominator space x end fraction equals 1 end cell row cell limit as x rightwards arrow 0 of fraction numerator tan space a x over denominator sin space b x end fraction end cell equals cell limit as x rightwards arrow 0 of fraction numerator sin space a x over denominator tan space b x end fraction equals a over b end cell row cell limit as x rightwards arrow 0 of fraction numerator sin space a x over denominator sin space b x end fraction end cell equals cell limit as x rightwards arrow 0 of fraction numerator tan space a x over denominator tan space b x end fraction equals a over b end cell end table$

Karena sin space 2 x equals 2 space sin space x space cos space x , maka nilai dari $limit as x rightwards arrow 0 of fraction numerator 7 space tan to the power of 4 space x over denominator 3 x squared space sin squared space x space cos space x end fraction$ adalah:

$table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as x rightwards arrow 0 of fraction numerator 7 space tan to the power of 4 space x over denominator 3 x squared space sin squared space x space cos space x end fraction end cell row blank equals cell limit as x rightwards arrow 0 of fraction numerator 7 times tan space x times tan space x times tan space x times tan space x over denominator 3 times x times x times sin space x times sin space x times begin display style c o s space x end style end fraction end cell row blank equals cell limit as x rightwards arrow 0 of 7 over 3 times fraction numerator tan space x over denominator x end fraction times fraction numerator tan space x over denominator x end fraction times fraction numerator tan space x over denominator sin space x end fraction times fraction numerator tan space x over denominator sin space x end fraction times fraction numerator 1 over denominator cos space x end fraction end cell row blank equals cell 7 over 3 times 1 times 1 times 1 times 1 times 1 over 1 end cell row blank equals cell 7 over 3 end cell end table$

Jadi, $table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as x rightwards arrow 0 of end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 7 space tan to the power of 4 space x over denominator 3 x squared space sin squared space x space cos space x end fraction end cell end table equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 7 over 3 end cell end table$ .