Reason for non-existence of the limit:

Subsequent are the reasons when  will not exist.

Reason 1:     when left and right inclinations of f(x) are not similar in the neighbourhood of x = a =>  ≠ 

For example            (when [.] shows the greatest integer function)

It is obvious from the figure that left inclination of [x] at x = 2 is 1 while the right inclination of [x] at x = 2 is 2. That  cannot exist.

Reason 2:     If f(x) is not described in the neighbourhood of x = a. For example sec^-1(cosx) Here f(x) is may described at x = 0

But in the left neighbourhood of 0, seems

 sec^-1(cosx)

when, similar as for right inclination, in that type also in the right neighbourhood of x=0, cosx<1. Therefore sec-1(cosx) is not defined.

Reason 3:     When f(x) doesn't have a exclusive inclinations. For example sin1/x we can have that -1 ≤ sinx ≤ 1, that defines for all non-zero values of x, sin 1/x could consider finite values. But when x becomes very closer to zero sin 1/x could erratically oscillate between -1 and + 1. It shows that sin 1/x  couldn't have general tendency for very short value of x. Therefore sin1/x may not exist.

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