Jason Farmer

2021-01-16

Find the limits: As $x\to \mathrm{\infty },\frac{-2x\left({x}^{2}+4\right)}{-7-7{x}^{4}}$

Lacey-May Snyder

The given function is $\frac{-2x\left({x}^{2}+4\right)}{\left(-7-7{x}^{4}}$
Evaluate the limits at the endpoints as shown below.
$\underset{x\to \mathrm{\infty }}{lim}\frac{-2x\left({x}^{2}+4\right)}{-7-7{x}^{4}}$
$=\underset{x->\mathrm{\infty }}{lim}\frac{{x}^{3}\left(-2-\frac{8}{{x}^{2}}\right)}{{x}^{4}\left(\frac{-7}{{x}^{4}}-7\right)}$
$=\underset{x->oo}{lim}\frac{{x}^{3}\left(-2-\frac{8}{{x}^{2}}\right)}{{x}^{4}\left(\frac{-7}{{x}^{4}}-7\right)}$
$=\underset{x\to \mathrm{\infty }}{lim}\frac{\left(-2-\frac{8}{{x}^{2}}\right)}{\left(x\left(\frac{-7}{{x}^{4}}-7\right)}=0$

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