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Math Help - Limit problem - help please guys

  1. #1
    Bar0n janvdl's Avatar
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    Limit problem - help please guys

    Hey guys,

    I've got a big test on friday and I was working out a past paper.

    I would appreciate it if you could show all the steps.

    The question is as follows:

    \mathop {\lim }\limits_{x \to 1} \frac{{x^{17}  - 1}}<br />
{{x - 1}}<br />

    There is a tip saying: "Think of a derivative".

    And the memo only says:

    f'(1) = 17

    I have no idea how they got that answer.


    Thanks in advance.
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  2. #2
    GAMMA Mathematics
    colby2152's Avatar
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    You may imply thinking of derivatives as applying L'Hopital's rule.
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  3. #3
    Senior Member Peritus's Avatar
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    you could also notice that:

    <br />
\frac{{x^{17}  - 1}}<br />
{{x - 1}} = x^{16}  + x^{15}  + x^{14}  +  \cdots  + x^2  + x + 1<br />

    and it's easy to see that the resulting function equals 17 when x approaches 1.
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  4. #4
    Super Member PaulRS's Avatar
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    Well there are many ways.

    I suppose that f(x)=x^{17}-1

    Right?

    That might suggest applying L'H˘pital's rule .

    Another Idea would be: \frac{x^{17}-1}{x-1}=1+x+...+x^{16} for all reals x except for 0

    So: \lim_{x\rightarrow{1}}\frac{x^{17}-1}{x-1}=\lim_{x\rightarrow{1}}{\left(1+x+...+x^{16}\rig  ht)}

    Or you can consider the change of variable e^{u}=x so x\rightarrow{1}, u\rightarrow{0}
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  5. #5
    Bar0n janvdl's Avatar
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    Quote Originally Posted by PaulRS View Post
    I suppose that f(x)=x^{17}-1

    Right?
    \frac{x^{17}-1}{x - 1}
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  6. #6
    Super Member wingless's Avatar
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    Maybe he tried to say, f(x) = x^{17}-1 and g(x) = x-1. Then was going to use,
    If \frac{f(a)}{g(a)} is in indeterminate form \left ( \frac{0}{0}, \frac{\infty}{\infty} \right )
    then
    \lim_{x\to a}\frac{f(x)}{g(x)} = \lim_{x\to a}\frac{f'(x)}{g'(x)}
    which is the description of l'hopital's rule.
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  7. #7
    Bar0n janvdl's Avatar
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    I get it now. Thanks for all the help and replies guys, truly appreciated!
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  8. #8
    GAMMA Mathematics
    colby2152's Avatar
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    Quote Originally Posted by janvdl View Post
    I get it now. Thanks for all the help and replies guys, truly appreciated!
    You are welcome!
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  9. #9
    Math Engineering Student
    Krizalid's Avatar
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    A related problem.
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