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In my calculus homework I came across a much faster* solution for one of the problems I was working on by looking at how WebAssign (online homework website) solves similar problems:

$$\lim \limits_{x \to 1} \dfrac{x(1-x^4)}{(x - 1)} = \lim \limits_{x \to 1} \dfrac{x(1-x)(1+x+x^2+x^3)}{x-1}$$

What is going on here, does this transformation have a name?


*this is from the problem: $$y = x - x^5$$ with the point, $P(1 , 0)$, where I couldn't find a way to solve using $$\lim \limits_{x \to a} \dfrac{f(x) - f(a)}{(x - a)}$$ so I used $$\lim \limits_{h \to 0} \dfrac{f(x + h) - f(x)}{h}$$ which required me to build a big polynomial and factor out one of the h variable)

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If you get $\dfrac 00$ in this kind of rational function limit, it means that the numerator and denominator have (at least) a common linear factor. And it will be $x-a$ where $a$ is the number which $x$ is approaching. So to factor you just divide the numerator and the denominator by $x-a$ (long division). Repeat if necessary. –  Git Gud Jun 9 at 2:37
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Is there a particular reason that you undid the formatting of the equation? In general, we try to avoid images as much as possible, since they're much harder to search for or use. –  user61527 Jun 9 at 2:44
    
@user61527, what do you mean by undid? I might have been working on the formulas I mentioned in my context, but as regards the main formula, I just took a screen cap from the homework solution. I can format it now that I've leaned the syntax for this site. Give me a minute. –  Matt Jun 9 at 2:48
    
Ok, we were probably editing simultaneously then. –  user61527 Jun 9 at 2:49

2 Answers 2

up vote 14 down vote accepted

They've factored the numerator, using the ever-so-useful identity

$$x^n - 1 = (x - 1)(x^{n - 1} + x^{n - 2} + \dots + x + 1)$$

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This identity is amazing, and not just for going from LHS to RHS--it also is handy if you want to find the roots of the equation $x^{n-1}+x^{n-2}+\cdots+x+1=0$. –  anorton Jun 9 at 2:46
    
Ah, thanks for this. It's been 10+ years since I had precalc in college, so it's sometimes hard to find the identity, theorem, or formula needed when I've long forgotten it. –  Matt Jun 9 at 3:06
    
Do you happen to know if this identity has a name? –  WChargin Jun 9 at 3:37
    
@WChargin I don't know of a specific name beyond "difference of $n$th powers." –  user61527 Jun 9 at 3:38
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@WChargin: I propose we name it "The Best Identity". –  Wug Jun 9 at 11:28

It is called factorization. You can read about it here: http://en.wikipedia.org/wiki/Factorization

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