$\underset{x\to +\infty }{\mathop{\lim }}\,\frac{{{x}^{2}}+\sqrt{{{x}^{4}}+{{x}^{2}}-1}}{3{{x}^{2}}+2x+5}=\underset{x\to +\infty }{\mathop{\lim }}\,\frac{{{x}^{2}}(1+\sqrt{1+\frac{1}{{{x}^{2}}}-\frac{1}{{{x}^{4}}}})}{{{x}^{2}}(3+\frac{2}{x}+\frac{5}{{{x}^{2}}})}=\frac{1+\sqrt{1+0-0}}{3+0+0}=\frac{2}{3}$