# f-x-d-dx-f-x-

Question Number 131206 by Study last updated on 02/Feb/21
$${f}\left({x}\right)=\infty \\$$$$\frac{{d}}{{dx}}{f}\left({x}\right)=? \\$$
Commented by mr W last updated on 02/Feb/21
$$\infty\:{is}\:{not}\:{a}\:{variable},\:{is}\:{not}\:{a}\:{constant}. \\$$$${f}\left({x}\right)=\infty\:{has}\:{no}\:{meaning}! \\$$$$\underset{{x}\rightarrow{a}} {\mathrm{lim}}{f}\left({x}\right)=\infty\:{has}\:{meaning}. \\$$
Commented by JDamian last updated on 02/Feb/21
$$\infty\:{is}\:{not}\:{a}\:{real}\:{number} \\$$$$\infty\:\notin\:\Re \\$$
Commented by mr W last updated on 02/Feb/21
$$\infty\:{is}\:{not}\:{a}\:{number}\:{at}\:{all}! \\$$
Commented by MJS_new last updated on 03/Feb/21
$$\mathrm{I}\:\mathrm{still}\:\mathrm{believe}\:\mathrm{that}\:\mathrm{if}\:{f}\left({x}\right)=\left[{term}\:{without}\:{x}\right] \\$$$$\Rightarrow\:\frac{{d}}{{dx}}\left[{f}\left({x}\right)\right]=\mathrm{0} \\$$$$\mathrm{example}: \\$$$${f}\left({x}\right)=\frac{\mathrm{1}}{{r}};\:{r}\in\mathbb{R}\:\Rightarrow\:\frac{{d}}{{dx}}\left[{f}\left({x}\right)\right]=\mathrm{0} \\$$$$\mathrm{we}\:\mathrm{have}\:\mathrm{the}\:\mathrm{limit} \\$$$${f}'\left({p}\right)=\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{f}\left({p}+{h}\right)−{f}\left({p}−{h}\right)}{\mathrm{2}{h}} \\$$$$\mathrm{in}\:\mathrm{this}\:\mathrm{case} \\$$$${f}'\left({p}\right)=\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\underset{{r}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\frac{\mathrm{1}}{{r}}−\frac{\mathrm{1}}{{r}}}{\mathrm{2}{h}}\:= \\$$$$=\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\underset{{r}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{0}}{\mathrm{2}{h}}\:=\mathrm{0} \\$$$$\mathrm{if}\:\mathrm{I}'\mathrm{m}\:\mathrm{wrong}\:\mathrm{please}\:\mathrm{prove} \\$$
Commented by mr W last updated on 03/Feb/21
$${yes},{f}\left({x}\right)\:{musn}'{t}\:{have}\:{term}\:{with}\:{x},\:{but} \\$$$${it}\:{must}\:{be}\:{validly}\:{defined}\:{and}\: \\$$$${represent}\:{values}. \\$$
Answered by prakash jain last updated on 03/Feb/21
$$\frac{{d}}{{dx}}\left(\infty\right)=\mathrm{0} \\$$