Pertanyaan

x → 0 lim x 3 1 + tan x − 1 + sin x = ....

$x \to 0 lim \frac{1 + tan x - 1 + sin x}{x ^{3}} = ....$

$- 1$
$- \frac{1}{4}$
0
$\frac{1}{4}$
$1$

A. Azriah

Master Teacher

Mahasiswa/Alumni Universitas Negeri Yogyakarta

Jawaban terverifikasi

Jawaban

jawaban yang benar adalah D.

Pembahasan

Ingat! tan x = cos x sin x 1 − cos x = 2 ⋅ sin 2 2 1 x x → 0 lim x sin x = 1 x → 0 lim b x sin a x = b a Bentuk sekawan a − b adalah a + b Penyelesaian: Oleh karena itu, jawaban yang benar adalah D.

Ingat!

$tan x = \frac{sin x}{cos x}$
$1 - cos x = 2 \cdot sin^{2} \frac{1}{2} x$
$x \to 0 lim \frac{sin x}{x} = 1$
$x \to 0 lim \frac{sin a x}{b x} = \frac{a}{b}$
Bentuk sekawan $a - b$ adalah $a + b$

Penyelesaian:

$begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell stack text lim end text with x rightwards arrow 0 below fraction numerator square root of 1 plus tan space x end root minus square root of 1 plus sin space x end root over denominator x cubed end fraction end cell row blank equals cell stack text lim end text with x rightwards arrow 0 below fraction numerator square root of 1 plus tan space x end root minus square root of 1 plus sin space x end root over denominator x cubed end fraction times fraction numerator square root of 1 plus tan space x end root plus square root of 1 plus sin space x end root over denominator square root of 1 plus tan space x end root plus square root of 1 plus sin space x end root end fraction end cell row blank equals cell stack text lim end text with x rightwards arrow 0 below fraction numerator left parenthesis 1 plus tan space x right parenthesis minus left parenthesis 1 plus sin space x right parenthesis over denominator x cubed left parenthesis square root of 1 plus tan space x end root plus square root of 1 plus sin space x end root right parenthesis end fraction end cell row blank equals cell stack text lim end text with x rightwards arrow 0 below fraction numerator tan space x minus sin space x over denominator x cubed left parenthesis square root of 1 plus tan space x end root plus square root of 1 plus sin space x end root right parenthesis end fraction end cell row blank equals cell stack text lim end text with x rightwards arrow 0 below fraction numerator fraction numerator text sin end text x over denominator text cos end text x end fraction minus sin space x over denominator x cubed left parenthesis square root of 1 plus tan space x end root plus square root of 1 plus sin space x end root right parenthesis end fraction end cell row blank equals cell stack text lim end text with x rightwards arrow 0 below fraction numerator fraction numerator text sin end text space x over denominator text cos end text space x end fraction minus fraction numerator sin space x times cos space x over denominator cos space x end fraction over denominator x cubed left parenthesis square root of 1 plus tan space x end root plus square root of 1 plus sin space x end root right parenthesis end fraction end cell row blank equals cell stack text lim end text with x rightwards arrow 0 below fraction numerator fraction numerator sin space x open parentheses 1 minus cos space x close parentheses over denominator cos space x end fraction over denominator x cubed open parentheses square root of 1 plus tan space x end root plus square root of 1 plus sin space x end root close parentheses end fraction end cell row blank equals cell stack text lim end text with x rightwards arrow 0 below fraction numerator sin space x open parentheses 2 times sin squared space 1 half x close parentheses over denominator x cubed cos space x space open parentheses square root of 1 plus tan space x end root plus square root of 1 plus sin space x end root close parentheses end fraction end cell row blank equals cell 2 stack blank text lim end text with x rightwards arrow 0 below fraction numerator sin space x over denominator x end fraction open parentheses stack text lim end text with x rightwards arrow 0 below blank fraction numerator s i n space 1 half x over denominator x end fraction close parentheses squared times stack text lim end text with x rightwards arrow 0 below fraction numerator 1 over denominator cos space x open parentheses square root of 1 plus tan space x end root plus square root of 1 plus sin space x end root close parentheses end fraction end cell row blank equals cell 2 left parenthesis 1 right parenthesis open parentheses 1 half close parentheses squared times fraction numerator 1 over denominator cos space left parenthesis 0 right parenthesis open parentheses square root of 1 plus tan space left parenthesis 0 right parenthesis end root plus square root of 1 plus sin space left parenthesis 0 right parenthesis end root close parentheses end fraction end cell row blank equals cell 2 times 1 fourth times fraction numerator 1 over denominator 1 open parentheses square root of 1 plus square root of 1 close parentheses end fraction end cell row blank equals cell 2 over 4 times 1 half end cell row blank equals cell 1 fourth end cell end table end style$

Oleh karena itu, jawaban yang benar adalah D.