Kaycee Roche

2021-11-06

Determine the one-sided limits numerically or graphically. If infinite, state whether the one-sided limits are ∞ or −∞, and describe the corresponding vertical asymptote.

$\underset{x\to -{2}^{\pm}}{lim}\frac{4{x}^{2}+7}{{x}^{3}+8}$

bahaistag

Skilled2021-11-07Added 100 answers

Given function is:

$f\left(x\right)=\frac{4{x}^{2}+7}{{x}^{3}+8}$

$\underset{x\to -{2}^{+}}{lim}\left(\frac{4{x}^{2}+7}{{x}^{3}+8}\right)=\frac{\underset{x\to -{2}^{+}}{lim}(4{x}^{2}+7)}{\underset{x\to -{2}^{+}}{lim}({x}^{3}+8)}$

Put, x=-2

$=\frac{4{(-2)}^{2}+7}{{(-2)}^{3}+8}$

$=\frac{4\times 4+7}{-8+8}$

$=\frac{23}{0}$

$=\mathrm{\infty}$

and

$\underset{x\to -{2}^{-}}{lim}\left(\frac{4{x}^{2}+7}{{x}^{3}+8}\right)=\frac{\underset{x\to -{2}^{-}}{lim}(4{x}^{2}+7)}{\underset{x\to -{2}^{-}}{lim}({x}^{3}+8)}$

Put, x=-2

$=\frac{4{(-2)}^{2}+7}{{(-2)}^{3}+8}$

$=\frac{4\times 4+7}{-8+8}$

$=\frac{23}{0}$

$=-\mathrm{\infty}$

$\therefore x=-2$ is vertical asymptote.

Put, x=-2

and

Put, x=-2

Find the local maximum and minimum values and saddle points of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function

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