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Question Number 48291 by Rio Michael last updated on 21/Nov/18

Find the exact value of 0^0

$${Find}\:{the}\:{exact}\:{value}\:{of}\:\mathrm{0}^{\mathrm{0}} \\ $$

Commented by maxmathsup by imad last updated on 21/Nov/18

0^0 =lim_(x→0^+ ) x^x =lim_(x→0^+ ) e^(xln(x)) =e^0 =1 because lim_(x→0^+ ) xln(x)=0 for that we take 0^0 =1 .

$$\mathrm{0}^{\mathrm{0}} ={lim}_{{x}\rightarrow\mathrm{0}^{+} } \:\:\:{x}^{{x}} ={lim}_{{x}\rightarrow\mathrm{0}^{+} } \:\:\:\:\:{e}^{{xln}\left({x}\right)} \:\:={e}^{\mathrm{0}} =\mathrm{1}\:\:{because}\:{lim}_{{x}\rightarrow\mathrm{0}^{+} } \:\:{xln}\left({x}\right)=\mathrm{0}\:{for} \\ $$$${that}\:{we}\:{take}\:\mathrm{0}^{\mathrm{0}} =\mathrm{1}\:. \\ $$

Answered by MJS last updated on 21/Nov/18

0^0 is not defined 1^(st) step a÷b=(a/b)=c ⇒ a=c×b ⇒ b≠0 because if b=0 we have a÷0=(a/0)=c ⇒ a=c×0=0 example: 5÷0=(5/0)=c ⇒ 5=c×0=0 which is wrong conclusion: a÷0=(a/0) is not defined ⇒ ⇒ a÷b=(a/b)=c for b≠0 2^(nd) step a^n =a×a×a×... (n times) 0^n =0×0×0×... =0 (a^m /a^n )=a^(m−n) ⇒ (a^n /a^n )=a^0 =1 but because of our above conclusion a≠0 ⇒ 0^0 is not defined but sometimes it makes sense to define it for some reasons. in these cases we are free to set the value. f(x)=(x/x)=1 ∀x≠0 ⇒ we can define f(0)=1 to keep the function continous. or f(x)=x^x . usually we want to give it the value of the limit.

$$\mathrm{0}^{\mathrm{0}} \:\mathrm{is}\:\mathrm{not}\:\mathrm{defined} \\ $$$$\mathrm{1}^{\mathrm{st}} \:\mathrm{step} \\ $$$${a}\boldsymbol{\div}{b}=\frac{{a}}{{b}}={c}\:\Rightarrow\:{a}={c}×{b} \\ $$$$\Rightarrow\:{b}\neq\mathrm{0}\:\mathrm{because}\:\mathrm{if}\:{b}=\mathrm{0}\:\mathrm{we}\:\mathrm{have} \\ $$$${a}\boldsymbol{\div}\mathrm{0}=\frac{{a}}{\mathrm{0}}={c}\:\Rightarrow\:{a}={c}×\mathrm{0}=\mathrm{0} \\ $$$$\mathrm{example}:\:\mathrm{5}\boldsymbol{\div}\mathrm{0}=\frac{\mathrm{5}}{\mathrm{0}}={c}\:\Rightarrow\:\mathrm{5}={c}×\mathrm{0}=\mathrm{0}\:\mathrm{which}\:\mathrm{is}\:\mathrm{wrong} \\ $$$$\mathrm{conclusion}:\:{a}\boldsymbol{\div}\mathrm{0}=\frac{{a}}{\mathrm{0}}\:\mathrm{is}\:\mathrm{not}\:\mathrm{defined}\:\Rightarrow \\ $$$$\Rightarrow\:{a}\boldsymbol{\div}{b}=\frac{{a}}{{b}}={c}\:\mathrm{for}\:{b}\neq\mathrm{0} \\ $$$$ \\ $$$$\mathrm{2}^{\mathrm{nd}} \:\mathrm{step} \\ $$$${a}^{{n}} ={a}×{a}×{a}×...\:\left({n}\:\mathrm{times}\right) \\ $$$$\mathrm{0}^{{n}} =\mathrm{0}×\mathrm{0}×\mathrm{0}×...\:=\mathrm{0} \\ $$$$\frac{{a}^{{m}} }{{a}^{{n}} }={a}^{{m}−{n}} \:\Rightarrow\:\frac{{a}^{{n}} }{{a}^{{n}} }={a}^{\mathrm{0}} =\mathrm{1}\:\mathrm{but}\:\mathrm{because}\:\mathrm{of}\:\mathrm{our} \\ $$$$\mathrm{above}\:\mathrm{conclusion}\:{a}\neq\mathrm{0}\:\Rightarrow\:\mathrm{0}^{\mathrm{0}} \:\mathrm{is}\:\mathrm{not}\:\mathrm{defined} \\ $$$$ \\ $$$$\mathrm{but}\:\mathrm{sometimes}\:\mathrm{it}\:\mathrm{makes}\:\mathrm{sense}\:\mathrm{to}\:\mathrm{define}\:\mathrm{it} \\ $$$$\mathrm{for}\:\mathrm{some}\:\mathrm{reasons}.\:\mathrm{in}\:\mathrm{these}\:\mathrm{cases}\:\mathrm{we}\:\mathrm{are}\:\mathrm{free} \\ $$$$\mathrm{to}\:\mathrm{set}\:\mathrm{the}\:\mathrm{value}.\:{f}\left({x}\right)=\frac{{x}}{{x}}=\mathrm{1}\:\forall{x}\neq\mathrm{0}\:\Rightarrow\:\mathrm{we} \\ $$$$\mathrm{can}\:\mathrm{define}\:{f}\left(\mathrm{0}\right)=\mathrm{1}\:\mathrm{to}\:\mathrm{keep}\:\mathrm{the}\:\mathrm{function} \\ $$$$\mathrm{continous}.\:\mathrm{or}\:{f}\left({x}\right)={x}^{{x}} .\:\mathrm{usually}\:\mathrm{we}\:\mathrm{want}\:\mathrm{to} \\ $$$$\mathrm{give}\:\mathrm{it}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{the}\:\mathrm{limit}. \\ $$

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