Why what a function approaches need not be what it equals

What to assume they know

• Approaching from the left and from the right, and when the two disagree — one-sided limits and the criterion for existence
• Factorising a difference of two squares and cancelling a common factor
• The domain of a function given by a formula, and why a denominator's zero is excluded
• Reading a graph with a point removed
• Radian measure, and the values of sin and cos at π/2 and at 0
• A relation is nothing more than a chosen part of the product — a function as a rule together with its domain

What they should be able to do

• State the rule that a limit is read off values near a point and never at it, and justify it from the definition of the one-sided limits
• Show that (x² − 4)/(x − 2) has the limit 4 at x = 2 although the function has no value there
• Classify each of the chapter's ten illustrations by which of value and limit exists at the point in question
• Read Table 12.4 through Table 12.12 as evidence, and say what a table of this kind can and cannot establish
• Exhibit a function whose value at a point differs from its limit there, and point to the two markers on its graph that record each
• Explain why the domain restriction in Illustration 8 makes only one one-sided reading available
• State what the chapter means by writing +∞ for a limit, and why that is not a claim that a limit exists in the sense of the Summary box
• Identify, for a given function and point, which of the four possible combinations of value and limit applies

Where it usually goes wrong

• "To find a limit, substitute the point." That happens to work in seven of the chapter's illustrations, which is why §12.3 checks them numerically rather than assuming it. It fails on p. 220's opening function, on Illustration 10 and on Illustration 9, all in the same section. Substitution is a theorem for particular families of function, proved later in §12.3.2 — not a definition.
• "If a function has no value at a point, it has no limit there." The opening function of §12.3 is the counterexample, and every derivative in the rest of the chapter is a limit of exactly this shape.
• "Cancelling (x − 2) changes the function." It changes it at one point only, and that point is the one the limit is not allowed to look at. The chapter prints the restriction x ≠ 2 into the definition for this reason.
• "A table of six values proves the limit." It is evidence, and the chapter is careful to call these deductions rather than proofs, always attaching the proviso that nothing abrupt happens in the untabulated gap. The proof arrives in §12.3.2.
• "Writing +∞ means the limit is a number called infinity." Illustration 8 is saying the outputs exceed any bound you name. The chapter itself sets that case aside as outside the course.
• "Illustration 8's function is undefined at 0, so it is the same case as the opening function." It is not. The opening function's nearby values aim at one number; Illustration 8's run away. Same missing value, opposite verdict.
• "The hollow circle is where the function is." It is where the function isn't. On Fig 12.7 the function is at the filled dot on the x-axis.

Questions to check understanding

• Evaluate a limit at a point where the function is undefined, by cancelling the common factor and stating the restriction under which the cancellation holds
• Given a piecewise function, compute both the limit at a point and the value at that point, and say whether they agree
• "Does the limit exist? Does f(a) exist? Are they equal?" as three separate marks on one function
• Read a printed graph and state the limit and the value at a marked point
• Explain why a one-sided domain permits only one of the two readings
• Tabulate a function near a point and state the limit the table indicates, together with the assumption the deduction rests on

Examples worth working on the board

Values marked verified are worked out here from the chapter's printed data; no answer key was consulted.

• The opening function (§12.3, p. 220). h(x) = (x² − 4)/(x − 2), with x ≠ 2 printed as part of the definition. The chapter asks the reader to evaluate h near 2 and see the values cluster around 4. Verified: for every x ≠ 2, (x² − 4)/(x − 2) = (x + 2)(x − 2)/(x − 2) = x + 2, so the limit at 2 is 4 while h(2) does not exist at all.
• Fig 12.2 (p. 220), read off the page image. The line y = h(x) drawn with a hollow circle where it would cross the height 4 above x = 2. Marked points: (−2, 0), (0, 2), (2, 0), and the height 4 on the y-axis with dashed guide lines to the hollow circle. The picture is the line y = x + 2 with one point lifted out.
• Illustration 1 (pp. 221–222). f(x) = x + 10 at x = 5. Table 12.4 gives f at 4.9, 4.95, 4.99, 4.995, 5.001, 5.01, 5.1 as 14.9, 14.95, 14.99, 14.995, 15.001, 15.01, 15.1. Limit 15; and f(5) = 15 too.
• Illustration 2 (pp. 222–223). f(x) = x³ at x = 1. Table 12.5 gives f at 0.9, 0.99, 0.999, 1.001, 1.01, 1.1 as 0.729, 0.970299, 0.997002999, 1.003003001, 1.030301, 1.331. Limit 1, and f(1) = 1. Verified: 0.999³ = 0.997002999 and 1.001³ = 1.003003001 — both exact, worth showing so students stop reading these as rounded.
• Illustration 3 (p. 223). f(x) = 3x at x = 2. Table 12.6 gives f at 1.9, 1.95, 1.99, 1.999, 2.001, 2.01, 2.1 as 5.7, 5.85, 5.97, 5.997, 6.003, 6.03, 6.3. Limit 6 = f(2). Graph: Fig 12.4.
• Illustration 4 (pp. 223–224). The constant function f(x) = 3 at x = 2. Limit 3, and the chapter observes the same holds at every real point.
• Illustration 5 (p. 224). f(x) = x² + x at x = 1. Table 12.7 gives f at 0.9, 0.99, 0.999, 1.01, 1.1, 1.2 as 1.71, 1.9701, 1.997001, 2.0301, 2.31, 2.64 — note this table has three entries below and three above, but the ones above start further out. Limit 2 = f(1). Fig 12.5 is the parabola with the point (1, 2) marked by dashed guides; the axes are ticked −2, −1, 1, 2, 3, 4, 5 horizontally and 1, 2, 3, 4 vertically.
• Illustration 6 (p. 225). f(x) = sin x at π/2, angle in radians. Table 12.8 gives f at π/2 − 0.1, π/2 − 0.01, π/2 + 0.01, π/2 + 0.1 as 0.9950, 0.9999, 0.9999, 0.9950. Limit 1 = f(π/2). Note the symmetry in the table — sin is level at its peak, and the two sides give identical readings.
• Illustration 7 (p. 225). f(x) = x + cos x at 0. Table 12.9 gives f at −0.1, −0.01, −0.001, 0.001, 0.01, 0.1 as 0.9850, 0.98995, 0.9989995, 1.0009995, 1.00995, 1.0950. Limit 1 = f(0).
• Illustration 8 (p. 226). f(x) = 1/x², with the domain given as the positive reals only. The chapter says plainly that approaching 0 from the left is not available here, because those inputs are outside the domain. Table 12.10 gives f at 1, 0.1, 0.01 and 10⁻ⁿ as 1, 100, 10000 and 10²ⁿ, with n any positive integer. The chapter writes the limit as +∞ and remarks that limits of this kind are outside the course.
• Illustration 10 (p. 227). f(x) = x + 2 for x ≠ 1, and f(1) = 0. Table 12.12 gives f at 0.9, 0.99, 0.999, 1.001, 1.01, 1.1 as 2.9, 2.99, 2.999, 3.001, 3.01, 3.1. Verified: both one-sided readings are 3, so the limit is 3 — while the value at the point is 0. This is the single cleanest instance in the chapter of value and limit both existing and disagreeing.
• Fig 12.7 (p. 227), read off the printed page. The line y = f(x) drawn through the marked crossings (−2, 0) and the height 2 on the y-axis, with a hollow circle at height 3 above x = 1 (dashed guides to both axes) and a filled dot at (1, 0) on the x-axis. Both markers appear in one picture, which is why this figure and not Fig 12.2 should carry section 7.
• The four-cell table (section 10, an added assembly from the chapter's material). Value present and limit present, splitting in two: they agree in Illustrations 1 to 7, and they differ in Illustration 10. Value absent, limit present — the opening function on p. 220. Value present, limit absent — Illustration 9, and the two-clause function on p. 221. Value absent and limit absent — Illustration 8, whose function has no value at 0 and whose outputs run away rather than settling. That last cell is easy to leave empty; the chapter fills it, and the grid is only complete with it in.

Figures to have open

• Fig 12.2 redrawn (p. 220): the line with a hollow circle at the height 4 above x = 2, and the marked crossings. Standard schematic.
• Fig 12.7 redrawn (p. 227): the line with a hollow circle at (1, 3) and a filled dot at (1, 0). This is the brief's central picture; both markers must appear in the same frame. Verified against the printed page.
• Fig 12.5 redrawn (p. 224): the parabola y = x² + x with (1, 2) picked out by dashed guides, axes ticked as printed.
• A four-cell grid (value present/absent × limit present/absent) that fills in over section 10, with the both-present cell dividing to hold Illustration 10 beside Illustrations 1 to 7.
• No photograph is needed.

Where this sits in the book

• NCERT Class XI Mathematics, Chapter 12 "Limits and Derivatives", §12.3 Limits, printed pp. 220–228. Illustrations 1 to 10 run pp. 221–227; the closing sentence stating that value and limit may differ is at the top of p. 228.
• Tables 12.4 (p. 222), 12.5 (p. 222), 12.6 (p. 223), 12.7 (p. 224), 12.8 and 12.9 (p. 225), 12.10 and 12.11 (p. 226), 12.12 (p. 227).
• Figs 12.2 (p. 220), 12.4 (p. 223), 12.5 (p. 224), 12.6 and 12.7 (p. 227).
• Printed stale cross-reference, confirmed on the page image. The sentence introducing the deduction under Table 12.9 on p. 225 calls it Table 13.9, while the caption directly above the table reads Table 12.9. A leftover from the pre-2022 edition, in which this was Chapter 13. Cite the caption.
• Deliberate cross-references outside this chapter: Illustrations 1, 2 and 4 send the reader to Chapter 2 for their graphs rather than redrawing them here, and Illustration 6's sine graph is named as a figure in Chapter 3. Illustration 3 is not one of them — its graph is printed right here, as Fig 12.4 on p. 223, and Illustration 5's is too, as Fig 12.5 on p. 224.
• Chapter Summary, p. 254, which states in one line that the value and the limit at a point need not be the same.