I am trying to figure the definition of the limit in an intuitive way , such that i don't need any formal definitions when i am asking myself a question about limits
Can non-monotone functions have a limit ? I mean , if x1 > x2 and f(x1) < f(x2) , and there is a value c that x approaches , and if x1 is closer to c than x2 , then the above assumption would mean that , if the function have a limit , than : f(x) doesn't approaches L as x approaches c because a further value of x to c would bring a closer distance of f(x) to L ( for the above example ).
However , i could't prove that non-monotone functions don't have a limit, and i don't think it is true since there isn't a theorem for this :)
Can somebody provide an explanation : Why the non-intuitive definition of limit doesn't work for non-monotone functions ?
Let consider for example
$$f(x)=\frac{\sin x}x$$
as $x \to \infty$, $f(x)$ oscillates but the limit exists of course and it is zero.
Therefore non monotonic functions may or may not have a limit.