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Question Number 95325 by ~blr237~ last updated on 24/May/20

 lim_(n→∞)     (1/n)HCF(20,n) = 0       ?

$$\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\:\:\:\frac{\mathrm{1}}{{n}}{HCF}\left(\mathrm{20},{n}\right)\:=\:\mathrm{0}\:\:\:\:\:\:\:? \\ $$

Commented by mr W last updated on 24/May/20

1≤HCF(20,n)≤20 for n≥20  (1/n)≤((HCF(20,n))/n)≤((20)/n)  lim_(n→∞) (1/n)≤lim_(n→∞) ((HCF(20,n))/n)≤lim_(n→∞) ((20)/n)  0≤lim_(n→∞) ((HCF(20,n))/n)≤0  ⇒lim_(n→∞) ((HCF(20,n))/n)=0

$$\mathrm{1}\leqslant{HCF}\left(\mathrm{20},{n}\right)\leqslant\mathrm{20}\:{for}\:{n}\geqslant\mathrm{20} \\ $$$$\frac{\mathrm{1}}{{n}}\leqslant\frac{{HCF}\left(\mathrm{20},{n}\right)}{{n}}\leqslant\frac{\mathrm{20}}{{n}} \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{1}}{{n}}\leqslant\underset{{n}\rightarrow\infty} {\mathrm{lim}}\frac{{HCF}\left(\mathrm{20},{n}\right)}{{n}}\leqslant\underset{{n}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{20}}{{n}} \\ $$$$\mathrm{0}\leqslant\underset{{n}\rightarrow\infty} {\mathrm{lim}}\frac{{HCF}\left(\mathrm{20},{n}\right)}{{n}}\leqslant\mathrm{0} \\ $$$$\Rightarrow\underset{{n}\rightarrow\infty} {\mathrm{lim}}\frac{{HCF}\left(\mathrm{20},{n}\right)}{{n}}=\mathrm{0} \\ $$

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