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Evaluate the following limit:$\underset{x\to -\mathrm{\infty }}{lim}\frac{8{x}^{4}+3{x}^{2}-x}{\sqrt{9{x}^{8}-2{x}^{5}+4}}$
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Solution:
$\underset{x\to -\mathrm{\infty }}{lim}\frac{8{x}^{4}+3{x}^{2}-x}{\sqrt{9{x}^{8}-2{x}^{5}+4}}\phantom{\rule{0ex}{0ex}}=\underset{x\to -\mathrm{\infty }}{lim}\frac{{x}^{4}\left(8+\frac{3}{{x}^{2}}-\frac{1}{{x}^{3}}\right)}{{x}^{4}\sqrt{\left(9-\frac{2}{{x}^{3}}+\frac{4}{{x}^{8}}\right)}}\phantom{\rule{0ex}{0ex}}=\underset{x\to -\mathrm{\infty }}{lim}\frac{\left(8+\frac{3}{{x}^{2}}-\frac{1}{{x}^{2}}\right)}{\sqrt{9-\frac{2}{{x}^{3}}+\frac{4}{{x}^{8}}}}\phantom{\rule{0ex}{0ex}}=\frac{8+0-0}{\sqrt{9-0+0}}\phantom{\rule{0ex}{0ex}}=\frac{8}{\sqrt{9}}=\frac{8}{3}$