when does a one-sided limit doesn't exist

[eng_boody]

Hi,

I I'd like to ask ...what value(s) does the one-sided limit have so that it could be said that it failed to exist
.....

 

[albbg]

Consider for instance a trigonometric function like sin(x):

lim sin(x)
x->inf

it's not possible to find a value, since the value of the function continue to oscillate between -1 and +1.
In a little bit more mathematical term is not possible, choosen an arbitrary "e" to find a number \[\delta\] so that for each x\[\in\](\[\delta\],+\[\infty\]) then f(x)\[\in\](k-e,k+e) where k is the value of the limit. For instance, if I choose e=0.5 I cannot fint any \[\delta\].
Another function (non trigonometric) can be

x-int(x)

 

[eng_boody]

I could imagine the example you gave me, but I have problem with the formal definition of cauchy that
contain the two inequalities , it will be helpful if you explained the interpretation of the definition,

I searched the internet for vids explaining that but I could understand it

I found that definition that I couldn't understand in Ron larson book, but actually am studying electrical engineering
and calculus therefore it is a first course involves analysis.. I want a book that extend to two-variable calculus

do you suggest another book that is comprehensive and easy
with no such definitions, I saw on the internet another authors
like stewart, many others , what you suggest ..thnx