AP® Calculus AB · Unit 1 · Topic 1.9
Connecting Multiple Representations of Limits
A limit should give the same answer whether you look at a graph, a table of values, or the algebra — because all three are describing the same approach toward the same point. Algebra is the only one that proves it exactly.
A limit is one fact about a function, so every way of looking at that function should point to the same fact. A graph shows the curve closing in on a height as x closes in on a value. A table shows the output numbers tightening in on that same height as the input numbers tighten in on that value. Algebra lets you simplify the expression and plug in directly, or at least see what it's approaching. If the picture, the table, and the algebra disagree, something's wrong with one of them — usually a table that wasn't zoomed in close enough, or a graph read sloppily.
Moving between representations is mostly about translation. A graph's limit is the y-value the curve approaches from both sides at a given x, even if there's a hole there. A table's limit is read by plugging in x-values closer and closer to the target from both directions and watching the outputs converge. The algebra usually means factoring, canceling, or rationalizing to remove whatever is causing a 0/0 problem, then substituting. For (x2-1)/(x-1) at x=1: the graph shows a line with a hole at (1,2), the table shows outputs sliding toward 2 from both sides, and the algebra factors to (x-1)(x+1)/(x-1) = x+1, which gives 2 directly.
This topic is the bridge between the informal limit ideas from earlier in the unit and the formal algebraic techniques coming next. Once a vertical asymptote is involved, all three representations change together: the graph shoots off toward infinity, the table's outputs grow without bound instead of settling, and the algebra shows a denominator heading to 0 while the numerator doesn't — signaling the limit does not exist as a finite number.
Key terms
- Limit
- The height a function's outputs approach as the input approaches some value, whether or not the function reaches it there.
- Removable discontinuity (hole)
- A point where a function is undefined or disagrees with its limit, visible as a gap in the graph.
- Factoring/canceling
- Rewriting an expression to remove a common factor causing a 0/0 form, revealing the true limit algebraically.
- Vertical asymptote
- A vertical line the graph approaches but never crosses, where the limit grows without bound instead of settling on a number.
Watch it explained
- How it worksHow do you move between the picture, the table, and the equation for the same limit?
- Worked exampleFind the limit of (x2 - 1)/(x - 1) at x = 1 three ways: picture, table, and algebra.
- Common mistakeWhy isn't a calculator picture ever an exact answer?
- Going furtherWhat happens to all three representations when the function has a vertical asymptote there?
Check yourself
End of topic.