Question

Consider the function f(x) = 5 ; for x < 0 0, for x = 0 5, for x > 0

Show that f has no limit as x approaches 0

Answer 1

$$f(x) = \begin{cases} 5 &\text{for } x<0 \\ 0 &\text{for } x=0 \\ 5 &\text{for } x>0 \\ \end{cases}$$

$$\lim\limits_{x\to 0^-}f(x)=5$$

$$\lim\limits_{x\to 0^+}f(x)=5$$

Since $\lim\limits_{x\to 0^-}f(x)=5=\lim\limits_{x\to 0^+}f(x),$ then $\lim\limits_{x\to 0}f(x)$ exists, and

$$\lim\limits_{x\to 0}f(x)=5$$