Question

lim (2x - 5) = 3
x->4

Prove this statement using the e, delta definition of a limit and illustrate with a diagram.

Answer 1

a diagram?

Answer 2

Thats just the problem, I think I can probably get the diagram honestly.

Answer 3

draw a line, put the coordinate \((4,3)\) on the line and a little interval round it

Answer 4

Okay. That part makes sense.

Answer 5

you need to find the \(\delta\) that works for any \(\epsilon\) so as usual you work backwards
start with the definition of the limit
\[\lim_{x\to 4}2x+5=3\] means:
given \(\epsilon>0\) there is a \(\delta\) such that if \(|x-4|<\delta\) you have \(|2x-5-3|<\epsilon\) and now it is algebra

Answer 6

typo there of course i meant
\[\lim_{x\to 4}2x-5=3\]

Answer 7

the algebra is not bad in this case
it is the idea of working backwards that is usually confusing

Answer 8

You nailed my problem. Exact definition of limits is killing me/

Answer 9

you want \[|2x-5-3|<\epsilon\] so we force it to work
\[|2x-5-3|=|2x-8|=|2(x-4)|=2|x-4|\] is a start

Answer 10

now you want to make sure that
\[2|x-4|<\epsilon\] but you control the size of \(|x-4|\) so at this point you can say "pick \(\delta=\frac{\epsilon}{2}\)"
that way we know if
\[|x-4|<\frac{\epsilon}{2}\] we are assured that
\[2|x-4|<\epsilon\]

Answer 11

then you run the algebra backwards, since each step is reversible, and you have your "proof"

in real life of course once you get the the step i wrote above you are done, but the proof is running the steps in reverse

Answer 12

I would just add...
in the definition we need \[0<|x-4|<\delta\]

Answer 13

not coincidentally if you have a line with slope \(m\) you can pick \(\delta=\frac{\epsilon}{m}\)

Answer 14

what @Zarkon said
he always keeps me honest

Answer 15

So the answer to exact definitions are generally rather abstract?

Answer 16

if you mean by abstract you have to use a general \(\epsilon\) rather than a number, like say \(0.01\) then yes
your job is to find the \(\delta\) that works for every \(\epsilon\) which in plain english means \(\delta\) is a function of \(\epsilon\)
in this example you can use \(\delta=\frac{\epsilon}{2}\)

Answer 17

Okay. That makes sense for the most part. Thanks!

Answer 18

practice will make it make more sense
write the definition down in the general case
then write what it means in your specific case
then figure out what \(\delta\) you need from algebra

at the beginning it should be obvious, as in the algebra i wrote above
gets harder later