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Question Number 82682 by zainal tanjung last updated on 23/Feb/20

$$\mathrm{Help}\mathrm{me}\mathrm{please}....!!$$$$\underset{\mathrm{x}\to 0}{\mathrm{Lim}}{\left(\frac{1}{\mathrm{ex}}\right)}^{6\mathrm{x}}=...$$

Commented by john santu last updated on 23/Feb/20

$$\underset{x\to 0}{\lim }{(1+\frac{1}{ex}-1)}^{6x}=\underset{x\to 0}{\lim }{(1+\frac{1-ex}{ex})}^{6x}$$$$=e{}^{\underset{x\to 0}{\lim }\left(\frac{1-ex}{ex}\right)\times 6x}=e^{\underset{x\to 0}{\lim }\left(\frac{6x-6{ex}^2}{ex}\right)}$$$$=e{}^{\underset{x\to 0}{\lim }\left(\frac{1-ex}{e}\right)}=e{}^{\frac{1}{e}}.$$

Commented by zainal tanjung last updated on 23/Feb/20

$$\underset{\mathrm{m}\to \mathrm{\infty }}{\lim }{[1+\frac{\mathrm{m}-\mathrm{e}}{\mathrm{e}}]}^{\frac{6}{\mathrm{m}}}$$$$\underset{\mathrm{m}\to \mathrm{\infty }}{\lim }{\left[{(1+\frac{1}{\left(\frac{\mathrm{e}}{\mathrm{m}-\mathrm{e}}\right)})}^{\left[\frac{\mathrm{e}}{\mathrm{m}-\mathrm{e}}\right]}\right]}^{\left[\frac{\mathrm{m}-\mathrm{e}}{\mathrm{e}}\right]\left(\frac{6}{\mathrm{m}}\right)}$$$${\mathrm{e}}^{\underset{\mathrm{m}\to \mathrm{\infty }}{\lim }\left[\frac{6\mathrm{m}-6\mathrm{e}}{\mathrm{me}}\right]}={\mathrm{e}}^{\frac{6}{\mathrm{e}}}=\sqrt[\mathrm{e}]{{\mathrm{e}}^6}$$$$$$

Commented by john santu last updated on 24/Feb/20

$$m?whereyougotm?$$

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