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Question Number 120515 by john santu last updated on 01/Nov/20

$$\underset{x\to \mathrm{\infty }}{\lim }{(3\sqrt[x]{2}-2.\sqrt[x]{3})}^x?$$

Answered by bramlexs22 last updated on 01/Nov/20

$$\ln \mathrm{L}=\underset{x\to \mathrm{\infty }}{\lim }\mathrm{x}\ln (3.2^{\frac{1}{\mathrm{x}}}-2.3^{\frac{1}{\mathrm{x}}})$$$$\mathrm{let}\frac{1}{\mathrm{x}}=\mathrm{t}\wedge \mathrm{t}\to 0^{+}$$$$\ln \mathrm{L}=\underset{\mathrm{t}\to 0^{+}}{\lim }\frac{\ln (3.2^{\mathrm{t}}-2.3^{\mathrm{t}})}{\mathrm{t}}$$$$\ln \mathrm{L}=\underset{\mathrm{t}\to 0^{+}}{\lim }\frac{3.\ln \left(2\right).2^{\mathrm{t}}-2.\ln \left(3\right).3^{\mathrm{t}}}{3.2^{\mathrm{t}}-2.3^{\mathrm{t}}}$$$$\ln \mathrm{L}=\ln \left(\frac{8}{9}\right).\mathrm{thus}\mathrm{the}\mathrm{limit}\mathrm{is}\mathrm{L}={\mathrm{e}}^{\ln \left(\frac{8}{9}\right)}=\frac{8}{9}$$

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