Limits from Graphs and Tables
- Read one-sided and two-sided limits from a graph, including where the function value differs
- Estimate a limit from a table of values and state what a table cannot establish
- Distinguish the three distinct reasons a limit fails to exist
The limit does not care about the point
This is the idea the whole unit rests on, and it is worth stating in its blunt form: lim f(x) as x → a has nothing to do with f(a). The limit describes where the outputs are heading as the inputs close in on a; the function value at a is a separate fact. So a graph can show an open circle at (2, 5) with a filled dot at (2, 1), and the limit as x → 2 is 5 while f(2) = 1. Both statements are true simultaneously. On a graph-reading question, trace along the curve toward a from each side and read the height the curve approaches, deliberately ignoring any dot sitting at x = a.
The two-sided limit exists only if the two one-sided limits agree
Write it as a rule you apply mechanically: lim(x→a) f(x) = L if and only if lim(x→a⁻) f(x) = L and lim(x→a⁺) f(x) = L. Approaching from the left means x values slightly less than a, so read the graph moving rightward toward a. Approaching from the right means x values slightly greater, so read moving leftward toward a. If those two heights differ, the two-sided limit does not exist — and saying "the limit is both 3 and 7" is not an answer. A piecewise-defined function is the standard vehicle for this, and the exam will hand you one whose two branches happen to disagree exactly at the breakpoint.
Three distinct ways a limit fails
Precision here separates a scored answer from a vague one. A limit fails to exist when: (1) the one-sided limits disagree — a jump, as in a piecewise function or a step function; (2) the outputs grow without bound — the function increases or decreases without bound, which is an infinite behavior and a vertical asymptote, and writing "the limit is ∞" is shorthand for "no finite limit exists"; or (3) the outputs oscillate without settling, the standard example being sin(1/x) as x → 0, which crosses every value in [−1, 1] infinitely often no matter how close you look. Naming which of the three is happening is frequently the actual question.
A table of values for f gives f(1.9) = 4.61, f(1.99) = 4.9601, f(1.999) = 4.996001, f(2.001) = 5.004001, f(2.01) = 5.0401, f(2.1) = 5.41. Estimate lim(x→2) f(x), and state what the table cannot tell you.
- 1.Read the left side: as x increases through 1.9, 1.99, 1.999, the outputs rise through 4.61, 4.9601, 4.996001 — heading toward 5.
- 2.Read the right side: as x decreases through 2.1, 2.01, 2.001, the outputs fall through 5.41, 5.0401, 5.004001 — also heading toward 5.
- 3.The two one-sided estimates agree, so the limit appears to be 5.
- 4.State the limitation: a table samples finitely many points, so it can never prove a limit. A function could behave normally at every tabulated value and do something else between them.
- 5.Also note what is absent: the table contains no value at x = 2, so it says nothing whatever about f(2), which may be 5, may be something else, or may be undefined.
A table suggests a limit; it never establishes one. If a free-response question asks you to justify a limit, use algebra or a theorem — never "the table shows it approaches 5." Conversely, if a question says estimate or approximate, the table is exactly what it wants.
A graph shows f approaching height 3 as x → 4 from the left, approaching height 3 from the right, and a filled dot at (4, 7). What is lim(x→4) f(x)?
For which reason does lim(x→0) sin(1/x) fail to exist?
Answer the 2 checkpoints as you read.
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