sengguan.com: Rumus Praktis Matematika

Hitunglah Limit Berikut :

$\lim_{x\rightarrow \infty }\sqrt{x^{2}+8x+3}-\sqrt{x^{2}+2x-5}$

Jawab :

$\lim_{x\rightarrow \infty }\sqrt{x^{2}+8x+3}-\sqrt{x^{2}+2x-5}$

$= \lim_{x\rightarrow \infty }\left ( \sqrt{x^{2}+8x+3}-\sqrt{x^{2}+2x-5} \right )\left ( \frac{\sqrt{x^{2}+8x+3}+\sqrt{x^{2}+2x-5}}{\sqrt{x^{2}+8x+3}+\sqrt{x^{2}+2x-5}} \right )$

$=\lim_{x\rightarrow \infty }\frac{x^{2}+8x+3-\left ( x^{2}+2x-5 \right )}{\sqrt{x^{2}+8x+3}+\sqrt{x^{2}+2x-5}}$

$=\lim_{x\rightarrow \infty }\frac{6x+8}{\sqrt{x^{2}+8x+3}+\sqrt{x^{2}+2x+5}}:\frac{x}{\sqrt{x^{2}}}$

$=\lim_{x\rightarrow \infty }\frac{6+\frac{8}{x}}{\sqrt{1+\frac{8}{x}+\frac{3}{x^{2}}}+\sqrt{1+\frac{2}{x}-\frac{5}{x^{2}}}}$

$=\frac{6+0}{\sqrt{1+0+0}+\sqrt{1+0-0}}=3$

Cara kedua
Sekarang kita gunakan rumus praktis matematika sebagai berikut
$\lim_{x \to \infty }\sqrt{ax^{2}+bx+c}-\sqrt{ax^{2}+px+q}=\frac{b-p}{2\sqrt{a}}$

$\lim_{x \to \infty }\sqrt{x^{2}+8x+3}-\sqrt{x^{2}+2x-5}=\frac{8-2}{2\sqrt{1}}=3$