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everyone. I have been thinking of this problem recently and am wondering if anyone is able to prove/disprove it:

Let $R$ be a rational function, that is let $R(x)=p(x)/q(x)$ where $p$ and $q$ are polynomials of degree > 0. Assume that at some point $x=a$, the function $R(x)$ is undefined (e.g. by division by zero): That is $R(a)$=undefined. Then $$\lim_{x\to a}R(x)=\lim_{x\to a}R^*(x),$$ where $R^*(x)=p'(x)/q'(x)$, where $p'(x)$ and $q'(x)$ are the derivatives of the polynomials of $p$ and $q$ respectively.

Is this equation always true?

Pianoman1234
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No, it's not:

$$\lim_{x\to0}\frac1x\;\;\;\text{doesn't exist, yet}\;\;\lim_{x\to0}\frac01=0$$

Added after comment below . Still false:

$$\lim_{x\to0}x\frac{x-1}x\;\;\;\text{again doesn't exist, yet}\;\;\lim_{x\to0}\frac11=1$$

DonAntonio
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  • Sorry, I meant to add that deg p,q>0 – Pianoman1234 Dec 30 '16 at 22:01
  • @DanielePilkington-Scimone Still false. Read addition. – DonAntonio Dec 30 '16 at 22:06
  • when you say lim(x to 0)1/x doesn't exist, isn't the limit equal to +infinity? – Pianoman1234 Dec 30 '16 at 22:09
  • But then, why does it work in this example: say p(x)=(x-a)(x+a) and q(x)=(x-a). Then, lim(x approach a)(p(x)/q(x))=lim(x approach a)(p'(x)/q'(x)). Assume a is positive integer. – Pianoman1234 Dec 30 '16 at 22:11
  • @DanielePilkington-Scimone No, when I say it doesn't exist I mean that: it is neither $;\infty;$ nor $;-\infty;$, as you can check evaluating the corresponding one-sided limits. – DonAntonio Dec 30 '16 at 22:13
  • Actually I just found out the answer myself. Thank you for helping anyway – Pianoman1234 Dec 30 '16 at 22:19
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    @DanielePilkington-Scimone "For helping anyway"? The above is the correct answer to what you asked, whether you got it from here or from somewhere else. There is no other "the answer" ... – DonAntonio Dec 30 '16 at 22:20
  • I was being kind by saying thank you for helping me as in you showed me a counterexample and then talked to me in the chat. Next time, do you want me to say 'thanks for not helping me at all... – Pianoman1234 Dec 30 '16 at 22:27