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February 18th, 2015, 06:48 PM  #1 
Senior Member Joined: Feb 2014 From: Louisiana Posts: 156 Thanks: 6 Math Focus: algebra and the calculus  How to justify the cancelling of variables in a rational expression?
For example, say we have $\displaystyle \frac{x^4(x  1)}{x^2}$. The function is undefined at 0, but if we cancel the x's, we get a new function that is defined at 0. So, in this case, we have $\displaystyle x^2(x  1)$, then $\displaystyle x^2(x  1)(1)$, and since $\displaystyle \frac{x^2}{x^2} = 1$, we then have $\displaystyle \frac{x^4(x  1)}{x^2}$. However, this is a new function, since the domain has changed to exclude x = 0. How is this justified? Why can we go about changing the function in that way? Specifically, when we evaluate limits, in the case where we have $\displaystyle \frac{x^4(x  1)}{x^2}$, how do we know that cancelling the x's will lead to the correct limit, since that is in effect the limit of the function $\displaystyle x^2(x  1)$ and not $\displaystyle \frac{x^4(x  1)}{x^2}$?
Last edited by skipjack; February 18th, 2015 at 10:25 PM. 
February 18th, 2015, 07:59 PM  #2  
Math Team Joined: Dec 2013 From: Colombia Posts: 6,876 Thanks: 2240 Math Focus: Mainly analysis and algebra  Quote:
Quote:
${x^2 \over x^2} = 1$ for all $x \ne 0$ so it is clear that it is heading towards 1 for $x=0$ too  this can be shown analytically too. We also have the formula $$\left.\begin{array}{c}\lim_{x \to a} f(x) = F \\ \lim_{x \to a} g(x) = G\end{array}\right\} \implies \lim_{x \to a} f(x)g(x) = FG$$ And thus $$\lim_{x \to 0} {x^4(x1) \over x^2} = \lim_{x \to 0} \left({x^2 \over x^2}x^2(x1)\right) = \left(\lim_{x \to 0} {x^2 \over x^2} \right)\bigg(\lim_{x \to 0} x^2(x1)\bigg) = 1 \cdot 0 = 0$$ Last edited by skipjack; February 18th, 2015 at 10:26 PM.  
February 18th, 2015, 10:22 PM  #3 
Global Moderator Joined: Dec 2006 Posts: 17,720 Thanks: 1359  
February 19th, 2015, 05:16 AM  #4 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 933 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms 
This example may help. Let $f(x)=x$ when $x\ne3$ and $f(3)=\text{hat}.$ What is $\lim_{x\to3}f(x)$? 

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cancelling, expression, justify, rational, variables 
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