x lim(√3x+1+√3x-1)^6+(√3x+1-√3x-1)^6(x+x^2-1)^6+(x-x^2-1)^6x^3

Question

$\underset{x	o \infty}{lim}\frac{(\sqrt{3x+1}+\sqrt{3x-1}{)}^{6}+(\sqrt{3x+1}-\sqrt{3x-1}{)}^{6}}{{(x+\sqrt{{x}^{2}-1})}^{6}+{(x-\sqrt{{x}^{2}-1})}^{6}}{x}^{3}$

Accepted Answer

$\underset{x	o \infty}{lim}\frac{(\sqrt{3x+1}+\sqrt{3x-1}{)}^{6}+(\sqrt{3x+1}-\sqrt{3x-1}{)}^{6}}{{(x+\sqrt{{x}^{2}-1})}^{6}+{(x-\sqrt{{x}^{2}-1})}^{6}}{x}^{3}$

$\begin{gathered} =\underset{x	o \infty}{lim}{x}^{3}	imes \frac{{x}^{3}{{(\sqrt{3+\frac{1}{x}}+\sqrt{3-\frac{1}{x}})}^{6}+{(\sqrt{3+\frac{1}{x}}-\sqrt{3-\frac{1}{x}})}^{6}}}{{x}^{6}{{(1+\sqrt{1-\frac{1}{{x}^{2}}})}^{6}+{(1-\sqrt{1-\frac{1}{{x}^{2}}})}^{6}}} \ =\frac{(2\sqrt{3}{)}^{6}+0}{{2}^{6}+0}={3}^{3}=27 \end{gathered}$

Solutions · 1