Epsilon delta to N & M Definition

[prasannaworld]

Okay for a simple finite limit: e.g.
lim (3x) = 3
x->1

in the end I say:

"Therefore for every |x - 3| < delta, there exists an epsilon such that |3x-3| < epsilon"

Hence I can make delta really really small and the y bounds of epsilon will constrain the limit.



So let's come to the example I saw in an article
lim (SQRT(x)) = INF
x-INF

Okay so:
x > N - x is greater than any positive integer
Match N with M^2
x > M^2
SQRT(x) > M

Okay so how will I make my statement?

 

[sutupidmath]

so you are trying to prove that:

\lim_{x\rightarrow \infty}\sqrt{x}=\infty right?

What we want to prove is that \forall N&gt;0, \exists M&gt;0 such that whenever

x&gt;M, \sqrt{x}&gt;N


By observation we have, as you pointed out:\sqrt{x}&gt;N=&gt; x&gt;N^2

so our statement would be

\forall N&gt;0, \exists M=N^2&gt;0 such that whenever

x&gt;M=N^2=&gt;\sqrt{x}&gt;N

 

[prasannaworld]

Yes. That is what I wanted. Still I think the best way for me to get this is convince myself by trying to prove a false limit (I obviously should not be able to...)