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Question Number 157171 by mathocean1 last updated on 20/Oct/21

calculate lim_(n→+∞) (((n+ln(n)+1)/((5+(√n))^2 )))

$${calculate}\:\underset{{n}\rightarrow+\infty} {{lim}}\left(\frac{{n}+{ln}\left({n}\right)+\mathrm{1}}{\left(\mathrm{5}+\sqrt{{n}}\right)^{\mathrm{2}} }\right) \\ $$

Answered by puissant last updated on 20/Oct/21

lim_(n→+∞) (((n+ln(n)+1)/((5+(√n))^2 )))=lim_(n→+∞) (((n(1+((ln(n))/n)+(1/n)))/(n((5/( (√n)))+1)^2 ))) =lim_(n→+∞) (((1+((ln(n))/n)+(1/n))/(((5/( (√n)))+1)^2 )))= 1..

$$\underset{{n}\rightarrow+\infty} {\mathrm{lim}}\left(\frac{{n}+{ln}\left({n}\right)+\mathrm{1}}{\left(\mathrm{5}+\sqrt{{n}}\right)^{\mathrm{2}} }\right)=\underset{{n}\rightarrow+\infty} {\mathrm{lim}}\left(\frac{{n}\left(\mathrm{1}+\frac{{ln}\left({n}\right)}{{n}}+\frac{\mathrm{1}}{{n}}\right)}{{n}\left(\frac{\mathrm{5}}{\:\sqrt{{n}}}+\mathrm{1}\right)^{\mathrm{2}} }\right) \\ $$$$=\underset{{n}\rightarrow+\infty} {\mathrm{lim}}\left(\frac{\mathrm{1}+\frac{{ln}\left({n}\right)}{{n}}+\frac{\mathrm{1}}{{n}}}{\left(\frac{\mathrm{5}}{\:\sqrt{{n}}}+\mathrm{1}\right)^{\mathrm{2}} }\right)=\:\mathrm{1}.. \\ $$

Commented by mathocean1 last updated on 22/Oct/21

thanks le puissant.

$${thanks}\:{le}\:{puissant}. \\ $$

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