One limit, four views AB & BC

Each view makes something different obvious:

1. Graphical. Trace each side; holes, jumps, and asymptotes are visible instantly. Weak on exact values.
2. Numerical. A two-sided table shows convergence digit by digit. Suggestive, never conclusive.
3. Algebraic. Substitution and rewriting give exact answers. Says nothing about why until you interpret it.
4. Verbal. "As inputs approach 2 from either side, outputs approach 5." Carries the meaning; carries no computation.

Fluency means starting anywhere and reaching the others. An exam question that hands you a table and asks for an interpretation is testing the numerical-to-verbal link specifically.

Algebraic to graphical. $\frac{x^2-4}{x-2}$ simplifies to $x+2$ for $x \ne 2$, so the graph is the line $y = x+2$ with a hole at $(2, 4)$. The cancelled factor is the hole.

Graphical to algebraic. A hole at $(3, 5)$ suggests a factor of $(x-3)$ cancelling top and bottom. An asymptote at $x = 3$ suggests a factor of $(x-3)$ surviving in the denominator alone.

Numerical to verbal. Outputs closing on 4 from both sides becomes "as $x$ approaches 2 from either side, $f(x)$ approaches 4", which is exactly $\lim_{x\to 2} f(x) = 4$.

Verbal to algebraic. "The function grows without bound as $x$ nears 1 from the right" becomes $\lim_{x\to 1^+} f(x) = \infty$, which predicts a factor of $(x-1)$ in a denominator.

The representations are redundant on purpose, and the redundancy is the tool. Three checks that catch nearly everything:

1. Algebra says 6, table says 6. Confirmed.
2. Algebra says the limit fails, graph shows a smooth curve. Recheck the algebra; you probably mishandled a form.
3. Table suggests 0, graph shows wild oscillation. Trust the graph; the table's inputs were unlucky.

A specific and common disagreement: a table reporting a limit where the graph shows a jump. That happens when the table sampled one side only, which the graph makes obvious at a glance.

Algebra is authoritative on exact values. A table cannot prove a limit is exactly $\frac{1}{3}$ rather than $0.3333$, and a graph cannot resolve that at all.

The graph is authoritative on qualitative behavior: which side runs where, whether a discontinuity is a hole or a jump, whether oscillation is present. Algebra can establish these too, but slower.

Neither is authoritative about the function value unless it is shown. A limit of 5 tells you nothing about $f(2)$, in any representation. That independence, from Topic 1.2, survives every translation.