Pertanyaan

x → 0 lim 3 sin 5 x + x 4 2 x 2 + 1 − 1 = .... (SBMPTN 2016)

$x \to 0 lim \frac{2 x ^{2} + 1 - 1}{3 sin ^{5} x + x ^{4}} = ....$

(SBMPTN 2016)

$0$
$\frac{2}{3}$
$\frac{3}{4}$
$\frac{1}{2}$
$1$

8 dari 10 siswa nilainya naik

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Klaim

A. Azriah

Master Teacher

Mahasiswa/Alumni Universitas Negeri Yogyakarta

Jawaban terverifikasi

Jawaban

jawaban yang benaradalah E.

jawaban yang benar adalah E.

Pembahasan

Berikut langkah-langkah untuk menyelesaikan soal: Mengalikan fungsi dengan bentuk sekawannya. Membagi fungsi dengan variabel berpangkat tertinggi. Menentukan nilai limit dengan menggunakan sifat limit trigonometri. Ingat! x → c lim k=k x → c lim x = c x → 0 lim sin x x = 1 Berikut penyelesaiannya berdasarkan langkah-langkah: Oleh karena itu, jawaban yang benaradalah E.

Berikut langkah-langkah untuk menyelesaikan soal:

Mengalikan fungsi dengan bentuk sekawannya.
Membagi fungsi dengan variabel berpangkat tertinggi.
Menentukan nilai limit dengan menggunakan sifat limit trigonometri.

Ingat!

$x \to c lim k=k$
$x \to c lim x = c$
$x \to 0 lim \frac{x}{sin x} = 1$

Berikut penyelesaiannya berdasarkan langkah-langkah:

$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 2 x squared plus 1 end root minus 1 over denominator square root of 3 space sin to the power of 5 space x plus x to the power of 4 end root end fraction end cell row blank equals cell stack text lim end text with x rightwards arrow 0 below fraction numerator square root of 2 x squared plus 1 end root minus 1 over denominator square root of 3 space sin to the power of 5 space x plus x to the power of 4 end root end fraction times fraction numerator square root of 2 x squared plus 1 end root plus 1 over denominator square root of 2 x squared plus 1 end root plus 1 end fraction end cell row blank equals cell stack text lim end text with x rightwards arrow 0 below space fraction numerator open parentheses 2 x squared plus 1 close parentheses minus 1 over denominator open parentheses square root of 3 space sin to the power of 5 space x plus x to the power of 4 end root close parentheses open parentheses square root of 2 x squared plus 1 end root plus 1 close parentheses end fraction end cell row blank equals cell stack text lim end text with x rightwards arrow 0 below space fraction numerator 2 x squared over denominator open parentheses square root of 3 space sin to the power of 5 space x plus x to the power of 4 end root close parentheses open parentheses square root of 2 x squared plus 1 end root plus 1 close parentheses end fraction end cell row blank equals cell stack text lim end text with x rightwards arrow 0 below fraction numerator 2 x squared over denominator open parentheses square root of 3 space sin to the power of 5 space x plus x to the power of 4 end root close parentheses end fraction times stack text lim end text with x rightwards arrow 0 below fraction numerator 1 over denominator open parentheses square root of 2 x squared plus 1 end root plus 1 close parentheses end fraction end cell row blank equals cell stack text lim end text with x rightwards arrow 0 below fraction numerator fraction numerator 2 x squared over denominator x squared end fraction over denominator fraction numerator open parentheses square root of 3 space sin to the power of 5 space x plus x to the power of 4 end root close parentheses over denominator x squared end fraction end fraction times stack text lim end text with x rightwards arrow 0 below fraction numerator 1 over denominator open parentheses square root of 2 x squared plus 1 end root plus 1 close parentheses end fraction end cell row blank equals cell stack text lim end text with x rightwards arrow 0 below fraction numerator 2 over denominator open parentheses square root of fraction numerator 3 space sin to the power of 5 space x over denominator x to the power of 4 end fraction plus x to the power of 4 over x to the power of 4 end root close parentheses end fraction times stack text lim end text with x rightwards arrow 0 below fraction numerator 1 over denominator open parentheses square root of 2 x squared plus 1 end root plus 1 close parentheses end fraction end cell row blank equals cell stack text lim end text with x rightwards arrow 0 below fraction numerator 2 over denominator open parentheses square root of fraction numerator 3 space sin space x to the power of 4 over denominator x to the power of 4 end fraction times sin space x plus 1 end root close parentheses end fraction times stack text lim end text with x rightwards arrow 0 below fraction numerator 1 over denominator open parentheses square root of 2 x squared plus 1 end root plus 1 close parentheses end fraction end cell row blank equals cell fraction numerator 2 over denominator open parentheses square root of stack text lim end text with x rightwards arrow 0 below open parentheses fraction numerator 3 space sin space x to the power of 4 over denominator x to the power of 4 end fraction close parentheses times stack text lim end text with x rightwards arrow 0 below sin space x plus stack text lim end text with x rightwards arrow 0 below 1 end root close parentheses end fraction times fraction numerator 1 over denominator open parentheses square root of 2 left parenthesis 0 right parenthesis squared plus 1 end root plus 1 close parentheses end fraction end cell row blank equals cell fraction numerator 2 over denominator open parentheses square root of 3 blank stack text lim end text with x rightwards arrow 0 below open parentheses fraction numerator sin space x to the power of 4 over denominator x to the power of 4 end fraction close parentheses times sin space 0 plus 1 end root close parentheses end fraction times fraction numerator 1 over denominator open parentheses square root of 0 plus 1 end root plus 1 close parentheses end fraction end cell row blank equals cell fraction numerator 2 over denominator square root of 3 times 1 times 0 plus 1 end root end fraction times fraction numerator 1 over denominator square root of 1 plus 1 end fraction end cell row blank equals cell space 2 over 1 times 1 half end cell row blank equals 1 end table$

Oleh karena itu, jawaban yang benar adalah E.