Originally Posted by

**lilaziz1** Hey everyone. I have a question on the definition of limits basically. First, the definition states:

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Let f be a function defined on some open interval that contains the number a, except possibly at a itself. Then we say taht the limit of f(x) as x approaches a is L, and we write

$\displaystyle \lim_{x\rightarrow a} f(x) = L $

if for every number $\displaystyle \epsilon > 0 $ there is a corresponding number $\displaystyle \delta > 0 $ such that

if $\displaystyle 0 < |x - a| < \delta $ then $\displaystyle |f(x) - L| < \epsilon$

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So the definition is basically saying that if the difference between x -a (whether x is approaching from the left/right since its abs value bar) is approaching 0, then the difference between f(x) - L is approaching 0. Is that correct?

Now an example:

Prove that $\displaystyle \lim_{x\rightarrow3} (4x - 5) = 7 $

Solution:

So we say: if

$\displaystyle 0 < |x - 3| < \delta$ then $\displaystyle |(4x - 5) - 7| < \epsilon$

$\displaystyle |4x - 12| < \epsilon$

$\displaystyle 4|x - 3| < \epsilon$

$\displaystyle 0 < |x - 3| < \frac{\epsilon}{4}$

$\displaystyle 0 < |x - 3| < \delta$

And that's the end of it. I don't see how that proves anything besides that delta and epsilon are proportional. How does this prove anything?