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Math Help - Limit as n tends to infinity of...

  1. #1
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    Limit as n tends to infinity of...

    Q: Find lim as n tends to infinity of

    n(a^(1/n)-1)

    where a>0.

    Could anyone give me a hint of where to start please?

    Thanks.
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  2. #2
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    Rewrite as \frac{{a^{\frac{1}{n}}  - 1}}{{\frac{1}{n}}} then proceed.
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  3. #3
    Super Member redsoxfan325's Avatar
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    L'Hopital's Rule will be helpful.
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  4. #4
    Member Abu-Khalil's Avatar
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    Let f(x)=a^x then \lim_{n\to\infty}\frac{a^{\frac{1}{n}}-1}{\frac{1}{n}}=\lim_{x\to 0}\frac{a^x-a^0}{x-0}=f'(0).
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  5. #5
    Member Abu-Khalil's Avatar
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    In general, f'(x)=\lim_{h\to 0}\frac{a^{x+h}-a^x}{h}=a^x\lim_{h\to 0}\frac{a^h-1}{h}. Now let t=a^h-1\Rightarrow h=\log_a(t+1)=\frac{\log(t+1)}{\log a} and when h\to 0, t\to 0 so \lim_{h\to 0}\frac{a^h-1}{h}=\lim_{t\to 0}\frac{\log a}{\frac{1}{t}\log(t+1)}=\log a.

    Finally, f'(x)=a^x\log a.
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