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In my book this is termed as a theorem and the proof given is as follows :-

$$\begin{align} \lim_{x \to a}\left(\frac{x^n - a^n}{x - a}\right) &=\lim_{x \to a}\left(\frac{(x - a)*(x^{x-1} + x^{n-2}*a + x^{n-3}*a^2 + x^{n-4}*a^3 + \cdots + x^1*a^{n-2} + a^{n-1})}{x - a}\right) \\ &=(a^{n-1} + a*a^{n-2} + \cdots + a^{n-1}) \\ &=(a^{n-1} + a^{n-1} + \cdots + a^{n-1}) \\ &=(n*a^{n-1}). \end{align}$$

Everything made sense to me except

$$x^n - a^n = (x - a)*(x^{x-1} + x^{n-2}*a + x^{n-3}*a^2 + x^{n-4}*a^3 + \cdots + x^1*a^{n-2} + a^{n-1})$$

Somebody please enlighten me on this topic.

Shaun
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3 Answers3

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No, this isn't a binomial expansion. This is just long division. To check that it makes sense, we can just expand it out:

\begin{align*} &(x - a)(x^{n - 1} + ax^{n - 2} + a^2x^{n - 3} + \cdots + a^{n - 2}x + a^{n - 1}) \\ &= x(x^{n - 1} + ax^{n - 2} + a^2x^{n - 3} + \cdots + a^{n - 1}) \\ &~~~~~~~~~~~~~~~~~- a(x^{n - 1} + ax^{n - 2} + \cdots + a^{n - 2}x + a^{n - 1}) \\ &= (x^n \color{red}{+ ax^{n - 1} + a^2x^{n - 2} + \cdots + a^{n - 1}x}) \\ &~~~~~~~+ (\color{red}{-ax^{n - 1} - a^2x^{n - 2} - \cdots - a^{n - 1}x} - a^n) \\ &= x^n - a^n \end{align*}

Adriano
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  • Ok i got it. I was over thinking it and just forgot the basics. Anyway is my expansion of $x^n - a^n$ correct ? –  Jun 04 '16 at 17:37
  • The correct expansion is the one given by the book, namely: $$ x^n - a^n = (x - a)(x^{n - 1} + ax^{n - 2} + a^2x^{n - 3} + \cdots + a^{n - 2}x + a^{n - 1}) = (x - a)\sum_{k=0}^{n-1} a^k x^{n - 1 - k} $$ – Adriano Jun 04 '16 at 17:41
  • Thanks i mixed the the expansion of $(x-a)^n$ and this one. I will just edit the question so that future readers don't get confused. I am a moron. –  Jun 04 '16 at 17:47
2

To adress your question. In the expression $$ (x - a)\cdot(x^{n-1} + x^{n-2}\cdot a + x^{n-3}\cdot a^2 + x^{n-4}\cdot a^3 + \cdots + x^1\cdot a^{n-2} + a^{n-1}) $$ you may just expand it as $$ x \cdot \left(x^{n-1} + x^{n-2}\cdot a + x^{n-3}\cdot a^2 + x^{n-4}\cdot a^3 + \cdots + x^1\cdot a^{n-2} + a^{n-1}\right) $$ giving

$$\left(x^n + x^{n-1}\cdot a + x^{n-2}\cdot a^2 + x^{n-3}\cdot a^3 + \cdots + x^2\cdot a^{n-2} + x\cdot a^{n-1}\right) $$

then substract $$ a\cdot(x^{n-1} + x^{n-2}\cdot a + x^{n-3}\cdot a^2 + x^{n-4}\cdot a^3 + \cdots + x^1\cdot a^{n-2} + a^{n-1}) $$ that is substract

$$ (x^{n-1}\cdot a + x^{n-2}\cdot a^2 + x^{n-3}\cdot a^3 + x^{n-4}\cdot a^4 + \cdots + x^1\cdot a^{n-1} + a^n) $$

then you can see that all terms cancel except

$$ x^n-a^n. $$

Olivier Oloa
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1

Hint: Use the definition of derivative of a function. The limit can be seen as the derivative of the function $f(x)=x^n$ at $x=a$

By definition $f'(a)=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}$

Kushal Bhuyan
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