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I intend to understand the following theorem:

Theorem If $\lim_{x\to a} f(x) = L$ and $f(x) = g(x) $ except at $a$ then $\lim_{x\to a} g(x) = L$.

I read the following statement

Theorems are usually important results which show how to make concepts solve problems or give major insights into the workings of the subject.

here. (I e-mailed the author but none of his e-mails seem to be working. )

Within this context I have the following questions

  1. What is this theorem highlighting? What does it actually mean? What to "see" in this theorem?

  2. How can I figure out which types of problems will be solved by the "thing" stated by this theorem?

  3. What is the "major insight" given by this theorem into the workings of the subject of limits?

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  • $\begingroup$ Just analyze by taking $f(x)=x+2$ and $g(x)=\frac{x^2-4}{x-2}$ $\endgroup$ – Ekaveera Kumar Sharma Nov 17 '15 at 11:24
  • $\begingroup$ It means that limit is not depending on value of function in that point where we are taking limit. That is, for $\lim_{x\to a} f(x)$ it is not important what it is $f(a)$ $\endgroup$ – Cortizol Nov 17 '15 at 11:26
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  1. The theorem states that the limit of a function at a point does NOT depend on the value of the function at that point.

  2. Consider a continuous function $f$ (take $f(x)=x$ as an example). Then you have for all $a \in \mathbb R$ $$\lim\limits_{x \to a} f(x) =f(a)$$ Now modify $f$ at point $a$ to get $g$ such that $g(a) = a+1$ and $g$ equal to $f$ elsewhere. The theorem states that still $$\lim\limits_{x \to a} g(x) =f(a)$$ However obviously $g$ is no more continuous!

  3. The major insight is that the existence of the limit of a function at a point DOES NOT imply that the function is continuous.

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