$\displaystyle\lim_{x\to\infty }x(\sqrt{9x^2+1}-3x)$ ifadesinin degeri nedir?

Question

lim x->sonsuz x($\sqrt{9x^2+1}$-3x)ifadesinin degeri nedir? cevap 1/6 

Answer 1

$$\lim\limits_{x\to \infty}x\left(\sqrt{9x^2+1}-3x\right)=\lim\limits_{x\to \infty}x\left(\sqrt{9x^2+1}-3x\right) \frac{\sqrt{9x^2+1}+3x}{\sqrt{9x^2+1}+3x}$$ $$=$$ $$\lim\limits_{x\to \infty}x\cdot \frac{9x^2+1-9x^2}{\sqrt{9x^2+1}+3x} =\lim\limits_{x\to \infty} \frac{x}{\sqrt{9x^2+1}+3x} =\lim\limits_{x\to \infty} \frac{x}{|3x|\sqrt{1+\frac{1}{9x^2}}+3x}$$ $$=$$ $$\lim\limits_{x\to \infty} \frac{x}{|3x|\sqrt{1+\frac{1}{9x^2}}+3x}=\lim\limits_{x\to \infty} \frac{x}{3x\left(\sqrt{1+\frac{1}{9x^2}}+1\right)}=\frac{1}{6}$$