Pertanyaan

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

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

N. Puspita

Master Teacher

Jawaban terverifikasi

Jawaban

.

$table attributes columnalign right center left columnspacing 0px end attributes row cell limit as h rightwards arrow 0 of fraction numerator tan left parenthesis x plus h right parenthesis minus tan space x over denominator h end fraction end cell equals cell 1 plus tan squared x end cell end table$ .

Pembahasan

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

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

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

Dari aturan di atas, maka diperoleh

$begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as h rightwards arrow 0 of fraction numerator tan left parenthesis x plus h right parenthesis minus tan space x over denominator h end fraction end cell equals cell limit as h rightwards arrow 0 of fraction numerator tan space left parenthesis x plus h minus x right parenthesis left parenthesis 1 plus tan left parenthesis x plus h right parenthesis space tan space x right parenthesis over denominator h end fraction end cell row blank equals cell limit as h rightwards arrow 0 of fraction numerator tan space h left parenthesis 1 plus tan left parenthesis x plus h right parenthesis space tan space x right parenthesis over denominator h end fraction end cell row blank equals cell limit as h rightwards arrow 0 of fraction numerator tan space h over denominator h end fraction times limit as h rightwards arrow 0 of left parenthesis 1 plus tan left parenthesis x plus h right parenthesis space tan space x right parenthesis end cell row blank equals cell 1 times left parenthesis 1 plus tan left parenthesis x plus 0 right parenthesis tan space x right parenthesis end cell row blank equals cell 1 plus tan space x space tan space x end cell row blank equals cell 1 plus tan squared x end cell end table end style$

Jadi, $table attributes columnalign right center left columnspacing 0px end attributes row cell limit as h rightwards arrow 0 of fraction numerator tan left parenthesis x plus h right parenthesis minus tan space x over denominator h end fraction end cell equals cell 1 plus tan squared x end cell end table$ .