Pertanyaan

x → ∞ lim 2 x + 4 x 6 x = …

$x \to \infty lim \frac{6 ^{x}}{2 ^{x} + 4 ^{x}} = \dots$

P. Anggrayni

Master Teacher

Jawaban terverifikasi

Jawaban

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$begin mathsize 14px style limit as x rightwards arrow infinity of fraction numerator 6 to the power of x over denominator 2 to the power of x plus 4 to the power of x end fraction equals infinity end style$ .

Pembahasan

Ingat bahwa untuk . Selanjutnya perhatikan perhitungan berikut: lim x → ∞ 2 x + 4 x 6 x = = = lim x → ∞ 2 x + 4 x 6 x × 6 x 1 6 x 1 lim x → ∞ 6 x 2 x + 6 x 4 x 6 x 6 x lim x → ∞ ( 3 1 ) x + ( 3 2 ) x 1 Gunakan sifat di atas pada penyebut sehingga diperoleh lim x → ∞ 2 x + 4 x 6 x = = = = ( 3 1 ) ∞ + ( 3 2 ) ∞ 1 0 + 0 1 0 1 ∞ Dengan demikian, .

Ingat bahwa untuk 0 less than a less than 1 comma space limit as x rightwards arrow infinity of a to the power of x equals a to the power of infinity equals 0 .

Selanjutnya perhatikan perhitungan berikut:

$lim_{x \to \infty} \frac{6 ^{x}}{2 ^{x} + 4 ^{x}} = = = lim_{x \to \infty} \frac{6 ^{x}}{2 ^{x} + 4 ^{x}} \times \frac{\frac{1}{6 ^{x}}}{\frac{1}{6 ^{x}}} lim_{x \to \infty} \frac{\frac{6 ^{x}}{6 ^{x}}}{\frac{2 ^{x}}{6 ^{x}} + \frac{4 ^{x}}{6 ^{x}}} lim_{x \to \infty} \frac{1}{( \frac{1}{3} ) ^{x} + ( \frac{2}{3} ) ^{x}}$

Gunakan sifat di atas pada penyebut sehingga diperoleh

$lim_{x \to \infty} \frac{6 ^{x}}{2 ^{x} + 4 ^{x}} = = = = \frac{1}{( \frac{1}{3} ) ^{\infty} + ( \frac{2}{3} ) ^{\infty}} \frac{1}{0 + 0} \frac{1}{0} \infty$

Dengan demikian, $begin mathsize 14px style limit as x rightwards arrow infinity of fraction numerator 6 to the power of x over denominator 2 to the power of x plus 4 to the power of x end fraction equals infinity end style$ .