lim_(x→∞) ((rx^5 −4x^3 +3)/(sx^5 −2x+1)) = (1/4) 2r+3s=?


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Question Number 171385 by cortano1 last updated on 14/Jun/22

$\lim_{x \to \infty} \frac{{r x}^{5} - 4 x^{3} + 3}{{s x}^{5} - 2 x + 1} = \frac{1}{4}$ $2 r + 3 s = ?$
	Answered by qaz last updated on 14/Jun/22

	$\underset{x \to \infty}{l i m} \frac{{r x}^{5} - 4 x^{3} + 3}{{s x}^{5} - 2 x + 1} = \frac{r}{s} = \frac{1}{4}$ $\Rightarrow 2 r + 3 s = ? ? ?$
	Answered by nurtani last updated on 14/Jun/22

	$14 ? ? ?$
	Answered by alephzero last updated on 14/Jun/22

	$\lim_{x \to \infty} \frac{{r x}^{5} - 4 x^{3} + 3}{{s x}^{5} - 2 x + 1} =$ $\lim_{x \to \infty} \frac{r 5 x^{4} - 12 x^{2}}{s 5 x^{4} - 2 x} =$ $\lim_{x \to \infty} \frac{x^{2} (r 5 x^{2} - 12)}{x (s 5 x^{3} - 2)} =$ $\lim_{x \to \infty} \frac{r 5 x^{3} - 12 x}{s 5 x^{3} - 2 x} =$ $\lim_{x \to \infty} \frac{r 15 x^{2}}{s 15 x^{2}} = \lim_{x \to \infty} \frac{r}{s} = \frac{r}{s} = \frac{1}{4}$ $\Rightarrow r = k \land s = 4 r = 4 k, k \in Z$ $\Rightarrow 2 r + 3 s = 2 k + 3 (4 k) = 2 k + 12 k =$ $= 14 k$
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