One sided limits

Question

My graph is shown below

Can someone explain why $\lim_{x \rightarrow 5^-} = 2$ and $\lim_{x \rightarrow 5^+} = 2$?

Regards

Answer 1

When you look at the graph around $x=5$, you see that three things are happening.

First of all, there is a hole in the graph at $x=5$ which means that the function is not continuous.

There is a black dot over $x=5$ at $y=1$ that is $g(5)=1$

As you approach $5$ from either side of $x=5$ the curve approaches the hole at $y=2$, and that is why both one sided limits are $2$

Answer 2

Recall that by the definition of limit we have

$$\forall \varepsilon>0 \quad \exists \delta>0 \quad \text{such that}\quad \color{red}{\forall x\neq5}\quad|x-5|<\delta \implies|f(x)-L|<\varepsilon$$

therefore it doesn’t matter for the value of the limit whether or not $f(5)$ exists and, if it exists, what is the value of $f(5)$ or, in other words whether or not the function is continuous at $x=5$, in any case the limit is $2$.