Answer to Question #273542 in Functional Analysis for raiz

Question

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

Answer 1

The statement is False.

Counterexample

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

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

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

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

We see that

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

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

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

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

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