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Math Help - Product Rule with limits

  1. #1
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    Product Rule with limits

    Let n\in N(natural) and a\in C(complex). Show that x^n \rightarrow a^n as a \rightarrow n. [Hint:use the product rule and induction on n]

    I am studying for an exam when i came across this problem in my notes. Any help would be appreciated.
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  2. #2
    Member Mauritzvdworm's Avatar
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    What about x, is there not more structure?

    In order to use induction we will need x^1\rightarrow a^1 for the initial step

    Then for the induction hypothesis we assume
    x^k\rightarrow a^k for all k\geq 1

    then
    x^{k+1}=x^kx\rightarrow a^kx
    however the induction hypothesis holds for all k\geq 1, so we have
    x^{k+1}\rightarrow a^{k+1}
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  3. #3
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    No, that's all that we are given with regards to x. But your explanation is very helpful. For some reason I couldn't out the induction process together with the product rule.
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