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Question Number 5637 by sanusihammed last updated on 23/May/16

Limit [((3^(n+1) − 5^(n+1) )/(3^n − 5^n ))] x→∞ Please help. Thanks

$${Limit}\:\:\:\left[\frac{\mathrm{3}^{{n}+\mathrm{1}} \:−\:\mathrm{5}^{{n}+\mathrm{1}} }{\mathrm{3}^{{n}} \:−\:\mathrm{5}^{{n}} }\right] \\ $$$${x}\rightarrow\infty \\ $$$$ \\ $$$$ \\ $$$${Please}\:{help}.\:{Thanks} \\ $$

Answered by Yozzii last updated on 23/May/16

lim_(x→∞) (((3^(x+1) −5^(x+1) )/(3^x −5^x )))=lim_(x→∞) ((((3/5))^x ×3−5)/(((3/5))^x −1)) =((lim_(x→∞) {3(3/5)^x −5})/(lim_(x→∞) {(3/5)^x −1})) =((3×0−5)/(0−1)) =5

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\left(\frac{\mathrm{3}^{{x}+\mathrm{1}} −\mathrm{5}^{{x}+\mathrm{1}} }{\mathrm{3}^{{x}} −\mathrm{5}^{{x}} }\right)=\underset{{x}\rightarrow\infty} {\mathrm{lim}}\frac{\left(\frac{\mathrm{3}}{\mathrm{5}}\right)^{{x}} ×\mathrm{3}−\mathrm{5}}{\left(\frac{\mathrm{3}}{\mathrm{5}}\right)^{{x}} −\mathrm{1}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\frac{\underset{{x}\rightarrow\infty} {\mathrm{lim}}\left\{\mathrm{3}\left(\mathrm{3}/\mathrm{5}\right)^{{x}} −\mathrm{5}\right\}}{\underset{{x}\rightarrow\infty} {\mathrm{lim}}\left\{\left(\mathrm{3}/\mathrm{5}\right)^{{x}} −\mathrm{1}\right\}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\frac{\mathrm{3}×\mathrm{0}−\mathrm{5}}{\mathrm{0}−\mathrm{1}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{5} \\ $$

Commented by Rasheed Soomro last updated on 23/May/16

NiCE!

$$\mathcal{N}{i}\mathcal{CE}! \\ $$

Commented by sanusihammed last updated on 23/May/16

Thanks so much for your time

$${Thanks}\:{so}\:{much}\:{for}\:{your}\:{time} \\ $$

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