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Math Help - Prove power of limits using Mathematical induction.

  1. #1
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    Question Prove power of limits using Mathematical induction.

    It has been a while that I've done math. induction and I'm having trouble with this challenge:

    Use mathematical induction to prove that if n is a positive integer and lim f(x) as x-->a = L ; the lim [f(x)]^n as x-->a = [L]^n

    How should this proof be approached?
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  2. #2
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by Arturo_026 View Post
    It has been a while that I've done math. induction and I'm having trouble with this challenge:

    Use mathematical induction to prove that if n is a positive integer and lim f(x) as x-->a = L ; the lim [f(x)]^n as x-->a = [L]^n

    How should this proof be approached?
    Step 1: Show it's true for n=1. We know it's true for n=1 because that fact is given to us.

    Step 2: Assume it's true for n=k. So assume \lim_{x\to a}f^k(x)=L^k.

    Step 3: Prove it's true for n=k+1. This is shown below:

    \lim_{x\to a}f^{k+1}(x) = \underbrace{\lim_{x\to a}\left(f^k(x)\cdot f(x)\right) = \left(\lim_{x\to a}f^k(x)\right)\left(\lim_{x\to a}f(x)\right)}_{Limit~Product~Rule} = L^k\cdot L = L^{k+1}

    So we're done.
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  3. #3
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    Quote Originally Posted by redsoxfan325 View Post
    Step 1: Show it's true for n=1. We know it's true for n=1 because that fact is given to us.

    Step 2: Assume it's true for n=k. So assume \lim_{x\to a}f^k(x)=L^k.

    Step 3: Prove it's true for n=k+1. This is shown below:

    \lim_{x\to a}f^{k+1}(x) = \underbrace{\lim_{x\to a}\left(f^k(x)\cdot f(x)\right) = \left(\lim_{x\to a}f^k(x)\right)\left(\lim_{x\to a}f(x)\right)}_{Limit~Product~Rule} = L^k\cdot L = L^{k+1}

    So we're done.
    Wow, i didn't imagine it'll be so short.
    Thank you so much.

    P.S. How do u get the limit notations in that format, I tried wolfram alpha but can.
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  4. #4
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by Arturo_026 View Post
    Wow, i didn't imagine it'll be so short.
    Thank you so much.

    P.S. How do u get the limit notations in that format, I tried wolfram alpha but can.
    Click on the math equations in my posts and a window will pop up showing you the code.

    \lim_{x\to a}f(x) is written as \lim_{x\to a}f(x). \int_a^b f(x)\,dx is written as \int_{a}^{b}f(x)dx. \frac{a}{b} is written as \frac{a}{b}. \sqrt{a} is \sqrt{a}. a^b is a^{b}. Surround the expressions with math tags. The close math tag is [/tex] and the opening one is [tex]. Or you can just highlight the expression and click the Sigma ( \Sigma) button on the toolbar.
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