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my question seem dull but I really want to know in providing if the f(x) continuous or differentiable at some point a . We must provide that

$$\lim_{x\to a}f(x) = f(a)$$

I want to know in this step. Do I must use $(\epsilon - \delta )$ to provide that the limit of $f(x)$ as $x$ approach $a$ is $f(a)$ or just using normal limit rules to simplify the function and find the limit

aukxn
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    what do you mean by "normal limit rules"? –  Oct 31 '14 at 02:41
  • @Gage The OP means the rules you use in calculus, when finding the limit of a function as $x$ approaches a certain value, such as simplifying or otherwise changing the expression to an equivalent one in order to calculate the limit. They are essentially asking this: When proving that $f$ is continuous at $x = a$, they want to show $\lim \limits_{x \to a} f(x) = f(a)$. Do they do this by finding $, \delta$ given $\epsilon$ in the $\epsilon-\delta$ definition? Or is it acceptable to somehow manipulating $f(x)$ to calculate the limit as $x \to a$? – layman Oct 31 '14 at 02:46
  • @MathIsHardNoItsNot part of why I was asking is that some (all?) of these "normal limit rules" (or the proof that the expressions really are equivalent) rely on $\epsilon$ - $\delta$ proofs to show they actually are equivalent. So it seemed weird to think of the two as mutually exclusive ways to attack the problem. –  Oct 31 '14 at 02:51
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    @Gage That's probably an area of confusion for the person asking the question. Maybe you should explain your thinking to them in an answer. I think it would answer their question. – layman Oct 31 '14 at 02:52
  • @MathIsHardNoItsNot I think it would be easier to do that if I (or someone else) had a concrete example from the person asking of what they are thinking of when they say "normal limit rules". That was part of why I asked in the first place. –  Oct 31 '14 at 02:54
  • @Gage We should be able to come up with an example. Here's one: If $, f(x) = \begin{cases} \frac{x^{2} - 9}{x - 3} & x \neq 3 \ 6 & x = 3 \end{cases}$, and you want to show that $f$ is continuous at $x = 3$. According to the OP, it should be OK to just calculate using calculus the $\lim \limits_{x \to 3} \frac{x^{2} - 9}{x - 3}$ to show it is equal to $f(3)$. Maybe this example will help you explain your idea? – layman Oct 31 '14 at 02:58

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I think in general, you don't need to use $\epsilon-\delta$ argument unless you are asked, say in an exam. Sometimes $f$ is very complicated, it's not convenient to use definition to verify the limit, in particular, it's hard to select a proper $\delta$ based on $\epsilon$. That's why we developed those limit rule so that we don't need to check by definition at each time. Moreover, it's faster to use the limit rule.

But before applying the limit rule, you must make sure the rule is correct (which requires use $\epsilon-\delta$ argument to prove it), and you also need to make sure the conditions of the rules are satisfied.

And sometimes, if you want to show the limit is not a certain number, $\epsilon-\delta$ argument is more useful.

John
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