when do limits not exist?

Infinity is NOT a LIMIT. A function that increases (+ infinity) or decreases (- infinity) has no LIMIT.

A function has a LIMIT if it approaches from both sides the same FINITE value.

In effect, it is "Like horseshoes and hand grenades" Close enough is close enough.

Re-read the definition of a LIMIT. What it says is: You tell me how close you want the function to be to the LIMIT ( your requirement can be a small as you like) and I can tell you what value to plug into the function that will give you a result as close or closer than you said you wanted me to be.

For INFINITY: Any number you can name, I can name one bigger. Thus, there is no LIMIT.

they do not exist unless you specify them to exist

taek an example
you have a function f(x)

but this function has a valid result ONLY if x is between -1 and +1

so you must specify f(x) lim(-1>x<1)

cos f(x) does not give a valid result outside these limits

If you DONT specify these limits then this implies f(x) is valid for all values of x

For 1/x - if you dont specity lim(x>0) this implies 1/0 is a valid result (= infinity)
If you dont want infinity to be a valid result then you specify 1/x lim(x>0)
( In the physical world you never want the result of a function to be infinity so its implied that you never want anything to be divided by zero.)

(this is the problem of the maths in the big bang
According to the big bang at time t=0 then the universe was at a point and therefore distance was zero
All the equations of gravity etc become infinite when d=0 and therefore break down
As a result everthing we know about it has a limit t>0)

y = sin(x) doesn't have a limit. It just goes back and forth between -1 and +1, no matter how large x is.

Also y = sin(1/x) doesn't have a limit at x = 0.