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Difference Between Limit of Function at x = a and f(a)

y = f(x)
82051.png exists but f(a) does not exist.
f(x) = 82045.png
The value of function at x = a is of the form 0/0 which is indeterminate, i.e., f(a) does not exist.
But 82039.png = 82033.png = 2a. Hence 82027.png exists.
82021.png does not exist but f(a) exists.
f(x) = [x], where [] represents greatest integer function
The value of function at x = n (n ∈ I) is n, i.e., f(n) = n.
But 82015.png = n – 1 and 82008.png.
Hence 82002.png does not exist.
81996.png and f(a) both exist and are equal.
f(x) =81990.png
The value of function at x = 0 is 0, i.e., f(0) = 0.
Also 81983.png = 0 and 81977.png = 0, i.e., 81971.png exists.
81965.png and f(a) both exist but are unequal.
The value of function at x = 3 is 3, i.e., f(3) = 3. Also 81953.png i.e. 81947.pngexists.
But 81941.png.

Thus for limit to exist at x = a it is not necessary that function is defined at that point.

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