Pertanyaan

Tentukanlah h → 0 lim h sin ( x + h ) − sin x

Tentukanlah $h \to 0 lim \frac{sin ( x + h ) - sin x}{h}$

E. Lestari

Master Teacher

Mahasiswa/Alumni Universitas Sebelas Maret

Jawaban terverifikasi

Jawaban

.

$limit as h rightwards arrow 0 of fraction numerator sin left parenthesis x plus h right parenthesis minus sin space x over denominator h end fraction equals cos space x$ .

Pembahasan

Ingat kembali rumus selisih sinus, sifat limit dan limit fungsi trigonometri berikut. Dari aturan di atas, maka diperoleh Jadi, .

Ingat kembali rumus selisih sinus, sifat limit dan limit fungsi trigonometri berikut.

$limit as x rightwards arrow 0 of fraction numerator sin space a x over denominator b x end fraction equals limit as x rightwards arrow 0 of fraction numerator space a x over denominator sin space b x end fraction equals a over b$

Dari aturan di atas, maka diperoleh

$begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as h rightwards arrow 0 of fraction numerator sin left parenthesis x plus h right parenthesis minus sin space x over denominator h end fraction end cell equals cell limit as h rightwards arrow 0 of fraction numerator 2 space cos 1 half open parentheses x plus h plus x close parentheses space sin begin display style 1 half end style open parentheses x plus h minus x close parentheses over denominator h end fraction end cell row blank equals cell limit as h rightwards arrow 0 of fraction numerator 2 space cos begin display style 1 half end style open parentheses 2 x plus h close parentheses space sin begin display style 1 half end style open parentheses h close parentheses over denominator h end fraction end cell row blank equals cell limit as h rightwards arrow 0 of fraction numerator 2 space cos space open parentheses x plus begin display style 1 half end style h close parentheses space sin begin display style 1 half end style h over denominator h end fraction end cell row blank equals cell open parentheses limit as h rightwards arrow 0 of 2 space cos space open parentheses x plus 1 half h close parentheses close parentheses times limit as h rightwards arrow 0 of fraction numerator space sin begin display style 1 half end style h over denominator h end fraction end cell row blank equals cell open parentheses 2 cos space open parentheses x plus 1 half open parentheses 0 close parentheses close parentheses close parentheses times 1 half end cell row blank equals cell open parentheses 2 space cos space open parentheses x plus 0 close parentheses close parentheses times 1 half end cell row blank equals cell 2 space cos space x times 1 half end cell row blank equals cell cos space x end cell end table end style$

Jadi, $limit as h rightwards arrow 0 of fraction numerator sin left parenthesis x plus h right parenthesis minus sin space x over denominator h end fraction equals cos space x$ .