Limit Existence: Finding a Function's Behavior

[Gurasees]

Finding a limit entails understanding how a function behaves near a particular value of x. So what do we mean when we say that a limit doesn't exist (in context to the upper statement)? (From what i studied, i noticed that limit exists only for those functions which have a discontinuity in the form of a hole.)

 

[andrewkirk]

I think your intuition is correct. If a function is discontinuous at a point but its limit exists at that point, it can be converted to a function that is continuous at that point by setting the value at that point equal to the limit. But if the limit does not exist, there is no simple fix like that which can convert it to a continuous function.

This excludes the terminology that is sometimes used where one says the limit of a function as it approaches a point is $\infty$. That is not a proper limit. The intuition only works for proper (ie finite) limits.