The statement is False.
Counterexample
Let f(x)=xsinx.
The function f(x) is not defined at x=0.
x→0−limf(x)=x→0−limxsinx=1
x→0+limf(x)=x→0+limxsinx=1 We see that
x→0−limf(x)=1=x→0+limf(x) Then x→0limf(x) exists, and
x→0limf(x)=1The function f(x)=xsinx has a removable discontinuity at x=0.
Therefore the statement "If a function f is not defined at x=a then the limit x→alimf(x) never exists" is false.
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