Connecting Multiple Representations of Limits AB & BC

A limit can be presented as a graph, a table, a formula, or a sentence, and fluency means moving freely among them. A cancelled factor in the algebra is a hole in the graph. A surviving denominator factor is an asymptote. A two-sided table closing on one value is the sentence "as $x$ approaches $a$ from either side, $f(x)$ approaches $L$". Each view makes something different easy: the graph shows behavior, the algebra gives exact values, the table shows convergence happening.

Because the views are redundant, disagreement between them is a reliable error signal, and the errors it catches are the unit's usual ones. A table that reports a limit where the graph shows a jump was sampled on one side only. An answer that reports the function value rather than the limit shows up as a mismatch between the plotted dot and the traced height. And a graph read too precisely, or a table stopped too early, will disagree with the algebra that settles the question exactly.