if-lim-x-f-x-g-x-1-does-lim-x-f-x-g-x-0-

Question Number 4158 by 123456 last updated on 30/Dec/15

$$\mathrm{if} \\ $$$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\frac{{f}\left({x}\right)}{{g}\left({x}\right)}=\mathrm{1} \\ $$$$\mathrm{does} \\ $$$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}{f}\left({x}\right)−{g}\left({x}\right)=\mathrm{0}? \\ $$

Commented by 123456 last updated on 30/Dec/15

$${f}\left({x}\right)−{g}\left({x}\right)={g}\left({x}\right)\left[\frac{{f}\left({x}\right)}{{g}\left({x}\right)}−\mathrm{1}\right] \\ $$

Commented by prakash jain last updated on 30/Dec/15

$${f}\left({x}\right)={x}^{\mathrm{2}} −{x} \\ $$$${g}\left({x}\right)={x}^{\mathrm{2}} \\ $$$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\frac{{f}\left({x}\right)}{{g}\left({x}\right)}=\mathrm{1} \\ $$$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}{f}\left({x}\right)−{g}\left({x}\right)\neq\mathrm{0} \\ $$

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