Estimating a limit numerically from a table of values

Computing $f(x)$ for $x$ values approaching 3 from the left, such as $2.9, 2.99, 2.999$, gives outputs $5.9, 5.99, 5.999$ — these values get closer and closer to $6$. Computing $f(x)$ for $x$ values approaching 3 from the right, such as $3.1, 3.01, 3.001$, gives outputs $6.1, 6.01, 6.001$, which also approach $6$.

Since the outputs approach the same number, $6$, from both sides as $x$ gets closer to $3$, we conclude the two-sided limit exists and equals $6$. This matches the algebraic fact that for $x \neq 3$, $\frac{x^2-9}{x-3} = \frac{(x-3)(x+3)}{x-3} = x+3$, which approaches $3+3=6$ as $x \to 3$, even though the original function is undefined exactly at $x=3$.

The function has no output value at $x = 3$ at all, since it is undefined there. The open circle drawn on the graph at that point signals exactly this: the point is excluded from the graph. Even though the outputs approach six as $x$ nears three from both sides, the function itself is simply not defined at $x = 3$. This is the key distinction between the limit and the function's actual value at that point. Therefore, $f(3)$ is undefined.