Question #273542

If a function f is not defined at x = a then the limit lim f(x) as x approaches a never exists. true or false


1
Expert's answer
2021-11-30T18:24:13-0500

The statement is False.

Counterexample

Let f(x)=sinxx.f(x)=\dfrac{\sin x}{x}.

The function f(x)f(x) is not defined at x=0.x=0.


limx0f(x)=limx0sinxx=1\lim\limits_{x\to 0^-}f(x)=\lim\limits_{x\to 0^-}\dfrac{\sin x}{x}=1

limx0+f(x)=limx0+sinxx=1\lim\limits_{x\to 0^+}f(x)=\lim\limits_{x\to 0^+}\dfrac{\sin x}{x}=1

We see that


limx0f(x)=1=limx0+f(x)\lim\limits_{x\to 0^-}f(x)=1=\lim\limits_{x\to 0^+}f(x)

Then limx0f(x)\lim\limits_{x\to 0}f(x) exists, and


limx0f(x)=1\lim\limits_{x\to 0}f(x)=1

The function f(x)=sinxxf(x)=\dfrac{\sin x}{x} has a removable discontinuity at x=0.x=0.


Therefore the statement "If a function ff is not defined at x=ax = a then the limit limxaf(x)\lim\limits_{x\to a}f(x) never exists" is false.



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