Does the limit exist if a function approaches a limit where it is discontinuous??

Question

If the function is discontinuous at $x=0$ and$f(0)=4.5$ . And as $x$ approaches $0$, the value of function nears $3$.

Does the limit in this case exist?

Imho ,it does and equals $3$ but one of my friend disagreed.

I'd be grateful if someone even hints at my error.

Thank you.

Answer 1

The limit exists, and is $3$. The fact that the limit is not the value of the function there is what tells you the function isn't continuous.

Answer 2

The limit exists and is 3. As I recall form my first year of calculus (Analysis?): If a function f is real-valued, then the limit of f at p is L if and only if both the right-handed limit and left-handed limit of f at p exist and are equal to L. $lim_{x\rightarrow0^+}f(x)=lim_{x\rightarrow 0^-}f(x)=3$