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Question Number 203650 by Davidtim last updated on 24/Jan/24

$$\underset{x\to 0}{\lim }{\left(\frac{2^x+3^x}{2}\right)}^{\frac{2}{x}}=?$$

Answered by witcher3 last updated on 25/Jan/24

$$$$$$\frac{2}{\mathrm{x}}\ln \left(\frac{2^{\mathrm{x}}+3^{\mathrm{x}}}{2}\right)$$$$2^{\mathrm{x}}={\mathrm{e}}^{\mathrm{xln}\left(2\right)}=1+\mathrm{xln}\left(2\right)+\mathrm{o}\left(\mathrm{x}\right)$$$$3^{\mathrm{x}}={\mathrm{e}}^{\mathrm{xln}\left(3\right)}=1+\mathrm{xln}\left(3\right)+\mathrm{o}\left(\mathrm{x}\right)$$$$2^{\mathrm{x}}+3^{\mathrm{x}}=2+\mathrm{xln}\left(6\right)+\mathrm{o}\left(\mathrm{x}\right)$$$$\frac{2}{\mathrm{x}}\ln \left(\frac{2^{\mathrm{x}}+3^{\mathrm{x}}}{2}\right)=\frac{2}{\mathrm{x}}\ln (1+\frac{\mathrm{x}}{2}\ln \left(6\right)+\mathrm{o}\left(\mathrm{x}\right))=\frac{2}{\mathrm{x}}\left(\frac{\mathrm{x}}{2}\ln \left(6\right)\right)+\mathrm{o}\left(1\right)$$$$\to \ln \left(6\right)$$$$\underset{x\to 0}{\lim }{\left(\frac{2^{\mathrm{x}}+3^{\mathrm{x}}}{2}\right)}^{\frac{2}{\mathrm{x}}}={\mathrm{e}}^{\ln \left(6\right)}=6$$

Answered by Calculusboy last updated on 26/Jan/24

$$\mathit{S}\mathit{o}\mathit{l}\mathit{u}\mathit{t}\mathit{i}\mathit{o}\mathit{n}:\mathit{b}\mathit{y}\mathit{s}\mathit{u}\mathit{b}\mathit{d}\mathit{i}\mathit{r}\mathit{e}\mathit{c}\mathit{t}\mathit{l}\mathit{y},\mathit{w}\mathit{e}\mathit{g}\mathit{e}\mathit{t}\left(1^{\mathrm{\infty }}\right)$$$$\mathit{l}\mathit{e}\mathit{t}\mathit{y}={\left(\frac{2^{\mathit{x}}+3^{\mathit{x}}}{2}\right)}^{\frac{2}{\mathit{x}}}\left(\mathit{a}\mathit{p}\mathit{p}\mathit{l}\mathit{y}\mathit{l}\mathit{o}\mathit{g}\mathit{t}\mathit{o}\mathit{b}\mathit{o}\mathit{t}\mathit{h}\mathit{s}\mathit{i}\mathit{d}\mathit{e}\mathit{s}\right)$$$$\mathit{l}\mathit{o}\mathit{g}\mathit{y}=\mathit{l}\mathit{i}\underset{\mathit{x}\to 0}{\mathit{m}\mathit{l}\mathit{o}\mathit{g}}{\left(\frac{2^{\mathit{x}}+3^{\mathit{x}}}{2}\right)}^{\frac{2}{\mathit{x}}}\Rightarrow \mathit{l}\mathit{o}\mathit{g}\mathit{y}=\mathit{l}\mathit{i}\underset{\mathit{x}\to 0}{\mathit{m}}\frac{2}{\mathit{x}}\mathit{l}\mathit{o}\mathit{g}\left(\frac{2^{\mathit{x}}+3^{\mathit{x}}}{2}\right)$$$$\mathit{l}\mathit{o}\mathit{g}\mathit{y}=\mathit{l}\mathit{i}\underset{\mathit{x}\to 0}{\mathit{m}}\frac{2\frac{\mathit{d}}{\mathit{d}\mathit{x}}\left[\mathit{l}\mathit{o}\mathit{g}\left(\frac{2^{\mathit{x}}+3^{\mathit{x}}}{2}\right)\right]}{\frac{\mathit{d}}{\mathit{d}\mathit{x}}\left(\mathit{x}\right)}\Rightarrow \mathit{l}\mathit{o}\mathit{g}\mathit{y}=\mathit{l}\mathit{i}\underset{\mathit{x}\to 0}{\mathit{m}}\frac{2\left\{\frac{\frac{1}{2}[2^{\mathit{x}}\mathit{I}\mathit{n}\left(2\right)+3^{\mathit{x}}\mathit{I}\mathit{n}\left(3\right)]}{\left(\frac{2^{\mathit{x}}+3^{\mathit{x}}}{2}\right)}\right\}}{1}$$$$\mathit{l}\mathit{o}\mathit{g}\mathit{y}=\mathit{l}\mathit{i}\underset{\mathit{x}\to 0}{\mathit{m}}\frac{2^{\mathit{x}}\mathit{I}\mathit{n}\left(2\right)+3^{\mathit{x}}\mathit{I}\mathit{n}\left(3\right)}{\frac{2^{\mathit{x}}+3^{\mathit{x}}}{2}}\Rightarrow \mathit{l}\mathit{o}\mathit{g}\mathit{y}=2\cdot \frac{\mathit{l}\mathit{i}\underset{\mathit{x}\to 0}{\mathit{m}}[2^{\mathit{x}}\mathit{I}\mathit{n}\left(2\right)+3^{\mathit{x}}\mathit{I}\mathit{n}\left(3\right)]}{\mathit{l}\mathit{i}\underset{\mathit{x}\to 0}{\mathit{m}}(2^{\mathit{x}}+3^{\mathit{x}})}$$$$\mathit{l}\mathit{o}\mathit{g}\mathit{y}=2\cdot \frac{[\mathit{I}\mathit{n}\left(2\right)+\mathit{I}\mathit{n}\left(3\right)]}{(1+1)}$$$$\mathit{l}\mathit{o}\mathit{g}\mathit{y}=\mathit{I}\mathit{n}\left(6\right)$$$$\mathit{y}={\mathit{e}}^{\mathit{I}\mathit{n}\left(6\right)}$$$$\mathit{y}=6$$$$\therefore \mathit{l}\mathit{i}\underset{\mathit{x}\to 0}{\mathit{m}}{\left(\frac{2^{\mathit{x}}+3^{\mathit{x}}}{2}\right)}^{\frac{2}{\mathit{x}}}=6$$

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