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Question Number 13011 by sandy_suhendra last updated on 10/May/17

$$\mathrm{li}\underset{\mathrm{x}\to \mathrm{\infty }}{\mathrm{m}}{[5^{\mathrm{x}}+5^{3\mathrm{x}}]}^{\frac{1}{\mathrm{x}}}=?$$$$\mathrm{please}\mathrm{help}$$

Answered by sma3l2996 last updated on 10/May/17

$$\underset{x\to +\mathrm{\infty }}{\lim }{(5^x+5^{3x})}^{\frac{1}{x}}=\underset{x\to +\mathrm{\infty }}{\lim }{\left(5^{3x}(5^{-2x}+1)\right)}^{\frac{1}{x}}={\underset{x\to +\mathrm{\infty }}{\lim 5}}^3{(1+5^{-2x})}^{\frac{1}{x}}$$$$wehave{\underset{x\to +\mathrm{\infty }}{\lim 5}}^{-2x}=0$$$$so\underset{x\to +\mathrm{\infty }}{\lim }{(5^x+5^{3x})}^{\frac{1}{x}}={\underset{x\to +\mathrm{\infty }}{\lim 5}}^3{(1+5^{-2x})}^{\frac{1}{x}}=5^3{(1+0)}^0=5^3=125$$

Commented by sandy_suhendra last updated on 11/May/17

$${\mathrm{thank}}^{\prime }\mathrm{s}\mathrm{for}\mathrm{your}\mathrm{kindness}$$

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