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Question Number 61934 by Cypher1207 last updated on 12/Jun/19

$$\mathrm{Answer}:0^0=?$$

Commented by Kunal12588 last updated on 12/Jun/19

$$\mathrm{I}\mathrm{believe}{0}^0=1$$

Commented by MJS last updated on 12/Jun/19

$$x^x=1$$$$x=1^{\frac{1}{x}}$$$$\underset{x\to 0^{+}}{\lim }\frac{1}{x}=+\mathrm{\infty }$$$$\underset{x\to 0^{-}}{\lim }\frac{1}{x}=-\mathrm{\infty }$$$${\underset{t\to \pm \mathrm{\infty }}{\lim 1}}^t=1$$$$\Rightarrow$$$$x=1$$$$\mathrm{but}x=0\Rightarrow 0^0\ne 1$$$$$$$$x^x=1$$$$x\ln x=0$$$$\Rightarrow x=0\vee x=1$$$$\mathrm{but}\mathrm{we}\mathrm{could}\mathrm{also}\mathrm{write}$$$$0^0=1$$$$0\ln 0=0$$$$\mathrm{now}\mathrm{divide}\mathrm{by}0$$$$\ln 0=\frac{0}{0}=1(\mathrm{believing}\frac{0}{0}=1)$$$$\ln 0=1$$$${\mathrm{e}}^{\ln 0}={\mathrm{e}}^1$$$$0=\mathrm{e}$$

Answered by MJS last updated on 12/Jun/19

$$0^0\mathrm{is}\mathrm{not}\mathrm{defined}$$$$\mathrm{for}x\ne 0:x^0=1\Rightarrow 0\ln x=\ln 0\Rightarrow 0=0\mathrm{true}$$$$[\ln x\in \mathbb{C}\mathrm{for}x<0\Rightarrow 0\ln x=0]$$$$\mathrm{for}x>0:0^x=0\Rightarrow 0=\sqrt[x]{0}\Rightarrow 0=0\mathrm{true}$$$$[\mathrm{for}x<0\mathrm{we}\mathrm{get}{\left(\frac{1}{0}\right)}^x\mathrm{which}{\mathrm{isn}}^{\prime }\mathrm{t}\mathrm{defined}\mathrm{too}]$$$$\underset{x\to 0}{\lim }{x}^0=1$$$$\underset{x\to 0^{+}}{\lim }{0}^x=0$$

Commented by MJS last updated on 12/Jun/19

$$\mathrm{but}\mathrm{working}\mathrm{on}\mathrm{a}\mathrm{special}\mathrm{function}\mathrm{it}\mathrm{can}\mathrm{be}$$$$\mathrm{useful}\mathrm{to}\mathrm{define}\mathrm{it}$$$$y=x^0\Rightarrow 0^0:=1$$$$y=0^x\Rightarrow 0^0:=0$$$$y=x^x\Rightarrow 0^0:=1$$$$\mathrm{similar}\mathrm{problems}\mathrm{occur}\mathrm{with}\frac{0}{0}$$$$y=\frac{0}{x}\Rightarrow \frac{0}{0}:=0$$$$y=\frac{x}{x}\Rightarrow \frac{0}{0}:=1$$

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