Estimating Limit Values from Tables AB & BC

A table estimates $\lim_{x\to a} f(x)$ by evaluating $f$ at inputs marching toward $a$ from both sides without ever reaching it. When both columns settle on the same value, that value is the estimate; when they settle on different values, the limit does not exist. Convergence looks like digits locking into place, and the point itself is often undefined, which is precisely why the table is useful.

Two errors dominate. The first is sampling badly: coming in from one side only, stopping at steps of $0.1$ where nothing has settled, or reading movement as convergence. The second is misreading an indeterminate form, treating $\frac{0}{0}$ as though it were a number or as proof that the limit fails. Three functions can all produce $\frac{0}{0}$ at the same input and have limits of 1, 0, and none, so the form decides nothing on its own. A table is also only evidence: it samples finitely many inputs and can miss oscillation entirely.