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Evaluate-the-given-limit-lim-n-8-1-n-1-16-1-n-1-




Question Number 203508 by Fridunatjan08 last updated on 20/Jan/24
Evaluate the given limit:  lim_(n→∞) (((8)^(1/n) −1)/( ((16))^(1/n) −1))
$${Evaluate}\:{the}\:{given}\:{limit}: \\ $$$$\underset{{n}\rightarrow\infty} {{lim}}\frac{\sqrt[{{n}}]{\mathrm{8}}−\mathrm{1}}{\:\sqrt[{{n}}]{\mathrm{16}}−\mathrm{1}} \\ $$
Answered by esmaeil last updated on 20/Jan/24
=Y  (1/n)=p→(n→∞→p→0)  →Y=lim_(p→0)  ((2^(3p) −1)/(2^(4p) −1))=^(hopital) lim_(p→0)  ((2^(3p) ×3ln2)/(2^(4p) ×4ln2))  =(3/4)
$$={Y} \\ $$$$\frac{\mathrm{1}}{{n}}={p}\rightarrow\left({n}\rightarrow\infty\rightarrow{p}\rightarrow\mathrm{0}\right) \\ $$$$\rightarrow{Y}=\underset{{p}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{2}^{\mathrm{3}{p}} −\mathrm{1}}{\mathrm{2}^{\mathrm{4}{p}} −\mathrm{1}}\overset{{hopital}} {=}\underset{{p}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{2}^{\mathrm{3}{p}} ×\mathrm{3}{ln}\mathrm{2}}{\mathrm{2}^{\mathrm{4}{p}} ×\mathrm{4}{ln}\mathrm{2}} \\ $$$$=\frac{\mathrm{3}}{\mathrm{4}} \\ $$

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