Estimating a limit numerically near a hole in the domain

As $x$ approaches $9$ from the left (e.g., $x = 8.9, 8.99, 8.999$), computing $g(x) = \frac{\sqrt{x}-3}{x-9}$ gives values that get closer and closer to $0.1\overline{6}$, i.e. approximately $0.1667$. Similarly, approaching from the right (e.g., $x = 9.1, 9.01, 9.001$) also produces values approaching about $0.1667$. Since the function is undefined exactly at $x=9$, we rely entirely on this trend from nearby values rather than trying to plug in $x=9$ directly.

Because both the left-hand and right-hand trends converge to the same numerical value, approximately $0.1667$, we conclude that the two-sided limit exists and equals this common value. This matches the algebraic value $1/6$ (obtainable by rationalizing $\sqrt{x}-3$, though that technique isn't required here), confirming $\lim_{x\to 9} g(x) \approx 1/6$.