1.9 Connecting Multiple Representations of Limits

Limits appear in many forms

A limit can be represented graphically (reading a curve's behavior near a point), numerically (a table of values approaching x = a), analytically (using algebra, factoring, or limit laws), and verbally (a written description). Being able to translate between these representations is essential — the same limit value can be found or verified in multiple ways.

Consistency across representations

If a limit exists, all representations must agree: the graph should show the function approaching the same y-value from both sides, the table should show values converging to that number, and the algebraic computation should confirm it. Discrepancies often reveal discontinuities, holes, or points where the limit doesn't exist (e.g., jump or infinite discontinuities).

Using tables and graphs together

Tables show trends but never *prove* a limit — values might approach a number then diverge outside the shown range. Graphs give visual intuition about left- and right-hand limits. Combining representations helps confirm whether a two-sided limit exists, and helps identify function behavior like asymptotes, removable discontinuities, or oscillation that a single representation might miss.

Key terms

Four representations of limits

Graphical, numerical (table), analytical (algebraic), and verbal descriptions.

Why can't a table alone prove a limit?

It only samples finite values; behavior outside the sampled range could differ, so it only suggests a limit's value.

Condition for a two-sided limit to exist

The left-hand limit must equal the right-hand limit at that point.

Removable discontinuity on a graph

A hole where the limit exists but f(a) is undefined or differs from the limit value.

Best representation for exact limit proof

Analytical (algebraic) methods, such as factoring or applying limit laws.

Try an AP-style question

f(x) = x² for x < 1 and f(x) = 2x for x ≥ 1. What is lim(x→1) f(x)?

1. 1
2. 3/2
3. Does not exist
4. 2