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Math Help - Limit problem

  1. #1
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    Limit problem

    So i started calculus and am currently stuck at the following problem.

    Q.Evaluate by algebraic techniques:
    \lim_{x\rightarrow a}\frac{x^{n}-a^{n}}{x^{m}-a^{m}}

    The answer provided by the book is:
    \frac{n}{m}x^{n-m}

    Thanks.
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  2. #2
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    Re: Limit problem

    The answer is correct..

    factorise both terms of the fraction EX: x^n-a^n=(x-a)(x^(n-1)+ax^(n-2)+...+a^(n-1)....
    do the same for the denominator ..simplify and then the limit when x goes to a is the one you mentioned ....
    Thanks from topsquark
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  3. #3
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    Re: Limit problem

    Quote Originally Posted by hacker804 View Post
    So i started calculus and am currently stuck at the following problem.

    Q.Evaluate by algebraic techniques:
    \lim_{x\rightarrow a}\frac{x^{n}-a^{n}}{x^{m}-a^{m}}

    The answer provided by the book is:
    \color{red}\frac{n}{m}x^{n-m}
    Reply #2 is correct in approach, but there is a slight error in the OP.

    The correct answer should be \frac{n}{m}a^{n-m}
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  4. #4
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    Re: Limit problem

    Yes it's a mistake in the book.Thanks for the help.
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  5. #5
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    Re: Limit problem

    Quote Originally Posted by hacker804 View Post
    So i started calculus and am currently stuck at the following problem.

    Q.Evaluate by algebraic techniques:
    \lim_{x\rightarrow a}\frac{x^{n}-a^{n}}{x^{m}-a^{m}}

    The answer provided by the book is:
    \frac{n}{m}x^{n-m}

    Thanks.
    Since the limit is of the indeterminate form \displaystyle \begin{align*} \frac{0}{0} \end{align*}, L'Hospital's Rule can be used.

    \displaystyle \begin{align*} \lim_{x \to a} \frac{x^n - a^n}{x^m - a^m} &= \lim_{x \to a}\frac{ \frac{d}{dx} \left( x^n - a^n \right) }{ \frac{d}{dx} \left( x^m - a^m \right) }  \\ &= \lim_{ x \to a} { \frac{ n\,x^{n - 1} }{ m \, x^{ m - 1} } } \\ &= \lim_{x \to a} \frac{n}{m}\, x^{ n - m} \\ &= \frac{n}{m}\, a^{n - m} \end{align*}
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