Question

Applying the epsilon-delta limit definition

Answer 1

I'll just state wikipedia's definition for my own reference.
\( \forall \varepsilon > 0\ \exists \ \delta > 0 : \forall x\ (0 < |x - c | < \delta \ \Rightarrow \ |f(x) - L| < \varepsilon)\)

Answer 2

ok

Answer 3

the example my textbook gives is the following:
"Show that the \(\lim_{x\rightarrow1}(5x-3)=2\)"

Answer 4

Then, they use substitutions with what is given
c=1,L=2.

Answer 5

So we show this statement to be true? \(0<|x-1|<\delta \rightarrow|f(x)-2|<\epsilon\)

Answer 6

yes..
the wiki definition can be elaborately expanded as below:
\[
\text{if:} \lim_{x\to a^-}f(x)=\lim_{x\to a+}f(x)\\
\text{then,}\lim_{x\to a}f(x)\quad\text{exists and}\\
\lim_{x\to a^-}f(x)=\lim_{x\to a+}f(x)=\lim_{x\to a}f(x)
\]

Answer 7

and \[\epsilon\to0\]

Answer 8

as \(x\to1\), \(\delta\to0\implies\epsilon\to0\)

Answer 9

okay. Then the book says "We find delta by working backwards from the epsilon-inequality:
\(|(5x-3)-2|=|5x-5|<\epsilon\)
\(|x-1|<\epsilon/5 \)

Answer 10

so they essentially manipulate the f(x)-L expression to arrive at the x-c expression?

Answer 11

that would be backward mapping... map "f(x)" to "x" given "L"

Answer 12

called image

Answer 13

it is very similar to the process of obtaining the vertical asymptote of a function.

Answer 14

What I don' get is the last part.
"thus we can take \(\delta=\epsilon/5\) If 0<|x-1|<delta=epsilon/5, then
\(|(5x-3)-2|=5|x-1|<5(\epsilon/5)=\epsilon\)

Answer 15

so \(\epsilon\) exists and hence the limit is true.

Also, could you tell me WHY they made that particular scalar multiple for \(\delta\)?

Answer 16

ooooh

Answer 17

does that 5 give us any hints as to the rate the limit converges? or does it just pop up conincidentally?

Answer 18

\[
0<|x-1|<\delta\\
\epsilon>|(5x-3)-2|=5|x-1|<5\delta\\
\]
now, for a certain "x",
\[
\epsilon=5\delta
\]

Answer 19

now, what you said about hint of "derivatives" is not wrong..
Leibnitz's approach in declaring \[\lim_{\delta\to0}{\epsilon\over\delta}={dy\over dx}\]
was not a fluke...
many stalwarts did not realize that connection till the former dared to state it.

Answer 20

hmm- this is a bit offtopic, but why is it that one can often treat a differential (and differential notation) as actual numbers, but they don't follow all rules? I heard somewhere that one can redefine differentials as another set of numbers, but there are rules that they dont' follow that real numbers do follow.

Answer 21

in fact, if you search into the "Antique stronghold" of "Project Gutenberg" and "Google Books", you will stumble into the rare Math bibles.
(I was very upset on the concept of eigen values and I found the history of development of matrices on Gutenberg)

Answer 22

I do not understand the last question

Answer 23

why dont you give it a shot for f(x)=x^2-1 in the above process...
you'd end up weith the "First principle of differentiation" :)

Answer 24

well, leibniz interpreted dy/dx as an infinitisimal change of y divide by an infinitismal change in x. You can do things with differential form like solving separable differential equations such as dy/dx=1/x
you rewrite in differential form to get dy=1/x dx or y=ln x (differential form rewriting is like treating the notation as a fraction?)
also, you could just directly integrate
Integral dy/dx dx=integral 1/x dx
integral dy=ln x
y=ln x
The notation is weird because it suggests that dy and dx are atual numbers, but everyone says they aren't.

what should I do with f(x)=x^2-1?

Answer 25

no, they are not. they are "rational" functions

Answer 26

you can play with rational numbers as long as you avoid "divisions with "0""

Answer 27

yes, but doesn't dx approach zero? I mean, youg et the indeterminant form 0/0 in all derivatives when evalating the difference quotient limit with direct substitution...

Answer 28

"approach", yes, "equal it?" no

you will still have some infinitesimally small \(\delta\), but never "0"
to avoid these problems, you use L'Hopital's Rule....

Answer 29

mm- I'ma go ahead and try some practice with this epsilon delta stuff- my main motivation is to use it for riemann sums, but I'll have to understand this first. Thanks for all the help!

Answer 30

surething

Answer 31

nice day.. started at 6am with an incredible integration problem here
ending with a good talk on limits.

Answer 32

xD was it the integral of sqrt(tan x) by any chance?

Answer 33

dont remember exactly.. it was a form of Beta function

Answer 34

oh wow... I got a while before I'll even remotely understand that...

Answer 35

i htink i was\[\int_0^1\frac{\mathrm{d}x}{\sqrt{1-x^m}}\]