Connecting Multiple Representations of Limits

A limit describes what a function does near a point, not necessarily at it. You can see that behavior three ways. A graph shows it as the height the curve approaches as you slide toward x = c from either side. A table shows it as the output values settling down as the input values get closer and closer to c. Algebra shows it as a computation — direct substitution, factoring, rationalizing — that predicts that same settling value without ever plotting or listing anything. All three are reading the same underlying fact, so a correct graph, a correct table, and correct algebra must always agree.

For a piecewise function, each representation has to respect the split. Take f(x) = x + 2 for x < 1 and f(x) = 3x for x > 1. Near x = 1 from the left, the graph follows the line x + 2 and climbs toward height 3; a table of x-values like 0.9, 0.99, 0.999 shows outputs creeping toward 3; the algebra is just plugging 1 into x + 2. From the right, everything points to 3x at x = 1, which is also 3. All three pieces of evidence land on 3, so the limit is 3 even if f(1) itself is never actually defined that way or is missing entirely.

When a graph and an algebra step seem to clash, don't assume the algebra wins automatically — algebra can silently drop a domain restriction, usually by cancelling a factor. This is exactly why the limit definition ignores the function's actual value at c: it lets all three representations describe the same approach behavior even when the point itself is a hole. That same approach-behavior reading is what you'll reuse to read instantaneous slope off a curve once derivatives start.