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

$$Findtheexactvalueof{0}^0$$

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

$$0^0={lim}_{x\to 0^{+}}{x}^x={lim}_{x\to 0^{+}}{e}^{xln\left(x\right)}=e^0=1because{lim}_{x\to 0^{+}}xln\left(x\right)=0for$$$$thatwetake{0}^0=1.$$

Answered by MJS last updated on 21/Nov/18

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

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