Approaching from the left and from the right, and when the two disagree

What to assume they know

• Averaging over shorter and shorter intervals to get a speed at an instant — averages over shrinking intervals, and a value trapped between two families
• Reading a piecewise definition: which formula applies on which part of the domain, and which clause owns the boundary point
• The modulus function |x| and its graph
• Plotting a step or broken graph with filled and hollow endpoint markers
• The one-output rule that promotes a relation to a function — the graph of a function as the set of its input–output pairs

What they should be able to do

• State what the left hand limit and the right hand limit of a function at a point are, each in terms of the values the function takes on one side only
• Compute both one-sided limits of a piecewise-defined function at the join
• State the criterion under which the two-sided limit exists, and apply it to decide existence
• Exhibit a function that is defined at a point and has no limit there, and explain why those two facts are compatible
• Read a graph's filled and hollow endpoint markers correctly, and say which marker records a function value and which records a one-sided limit
• Compute both one-sided limits of |x|/x and of x/|x| at 0 and state that neither function has a limit there
• Given a piecewise function, determine every point at which its limit exists
• Explain why a table of values on one side alone can never settle a limit

Where it usually goes wrong

• "If f(a) exists, the limit at a exists." Fig 12.3, Fig 12.6 and Exercise 12.1 q25 all refute this directly. Every one of those functions has a stated value at the point and no limit there.
• "If the two clauses are different formulas, the limit must fail." Exercise 12.1 q23 at x = 0 has 2x + 3 on one side and 3(x + 1) on the other and the limit exists, because both formulas head for 3. The test is on the two values, never on the two expressions.
• "A hollow circle means the function is undefined there." In Fig 12.6 the hollow circle at (0, 2) sits directly above a filled dot at the origin. The function is defined at 0; it just does not take the value 2 there. The hollow circle records a limit, the filled dot a value.
• "You can settle a limit by tabulating values on one side." That table settles one one-sided limit, which is half a verdict. Illustration 9's table is deliberately printed with three entries on each side for this reason.
• "No limit means the function misbehaves badly." Each function here is simple and completely specified. The failure is only that two well-defined readings differ.
• "|x|/x and x/|x| must behave differently." They are equal wherever both are defined, since each is ±1 and they take the same sign. Students expect the inversion to matter and it does not.

Questions to check understanding

• Compute both one-sided limits of a two- or three-clause function at its join and state whether the limit exists
• "Find the limit, if it exists" on a modulus quotient at 0
• Given a piecewise function, list every point at which the limit exists
• Determine unknown constants in a piecewise definition so that a limit exists or matches a stated value at the join — the standard follow-on, set as Exercise 12.1 q28 and q32 (pp. 238–239)
• Sketch a function that is defined at a point but has no limit there
• One-mark: state the condition under which the two-sided limit is said to exist

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 chapter's counterexample (§12.3, p. 221). f is defined by two clauses: f(x) = 1 when x ≤ 0, and f(x) = 2 when x > 0. Note which clause owns the boundary — the inclusive sign is on the first clause, so f(0) = 1. Read off the page image; the extracted text of this page does render the sign correctly, but several inequality signs elsewhere in this book do not, so the image is the authority. The chapter records the left hand limit at 0 as 1 and the right hand limit as 2, and concludes that the limit does not exist.
• Fig 12.3 (p. 221), confirmed on the printed page. Two horizontal segments. The lower one sits at height 1 and runs leftward from the y-axis, with a filled dot at (0, 1). The upper one sits at height 2 and runs rightward, with a hollow circle at (0, 2). Only the labels (0, 1), (0, 2), y = f(x) and the four axis letters appear. The two markers carry the whole argument: the filled dot is the function's value, the hollow circle is a height the right hand limit approaches and the function never takes.
• Illustration 9 (§12.3, pp. 226–227). f(x) = x − 2 for x < 0, f(0) = 0, and f(x) = x + 2 for x > 0. Table 12.11 gives f at −0.1, −0.01, −0.001, 0.001, 0.01, 0.1 as −2.1, −2.01, −2.001, 2.001, 2.01, 2.1. Verified: every entry is the relevant clause evaluated — e.g. −0.001 − 2 = −2.001 and 0.001 + 2 = 2.001. Left hand limit −2, right hand limit 2, so no limit at 0; but f(0) is defined and equals 0, a value neither one-sided limit is anywhere near.
• Fig 12.6 (p. 227), read off the printed page. A line of slope 1 broken at the y-axis: hollow circles at (0, 2) and (0, −2), and a filled dot at the origin. Three marked points, one filled. This is the cleanest picture in the chapter of "defined here, no limit here" — the printed dot is the value 0, and it sits in the gap between the two hollow circles.
• Exercise 12.1 q25 (p. 238). f(x) = |x|/x for x ≠ 0, and f(0) = 0; find the limit as x → 0. Verified: for x > 0 the quotient is x/x = 1; for x < 0 it is −x/x = −1. Right hand limit 1, left hand limit −1, so no limit at 0. The stated value at 0 changes nothing.
• Exercise 12.1 q26 (p. 238). f(x) = x/|x| for x ≠ 0, f(0) = 0. Verified: the same two one-sided values, 1 and −1, for the same reason — the quotient is its own reciprocal here.
• Exercise 12.1 q30 (p. 239). f(x) = |x| + 1 for x < 0, f(0) = 0, and f(x) = |x| − 1 for x > 0; the question asks for which values of a the limit at a exists. The modulus bars in this item do not survive extraction — they are in the page image and absent from p239.txt. Take this item from the image only. Verified: as x → 0 from below, |x| + 1 → 1; from above, |x| − 1 → −1; so no limit at 0. At any a < 0 the function agrees with −x + 1 near a, and at any a > 0 with x − 1 near a, so the limit exists and equals the formula's value. The set of good a is every real number other than 0.
• Exercise 12.1 q23 (p. 238). f(x) = 2x + 3 for x ≤ 0 and f(x) = 3(x + 1) for x > 0; the limits at 0 and at 1 are both asked for. Verified: at 0 the left reading is 3 and the right reading is 3, so the limit is 3 even though the two clauses are different formulas — agreement, not sameness, is the test. At 1 only the second clause is in play near the point, giving 6.
• Exercise 12.1 q24 (p. 238). f(x) = x² − 1 for x ≤ 1 and f(x) = −x² − 1 for x > 1; the limit at 1 is asked for. Verified: left reading 0, right reading −2, so no limit. Note this pair against q23: same shape of question, opposite verdict. Run them side by side.

Figures to have open

• Fig 12.3 redrawn (p. 221): two horizontal rays, one filled and one hollow endpoint on the y-axis. Standard schematic; the marker styles are the content and must be exact.
• Fig 12.6 redrawn (p. 227): the broken line with hollow circles at (0, 2) and (0, −2) and a filled dot at the origin. Standard schematic, verified against the printed page.
• A two-column verdict card — left reading, right reading, verdict — reusable across every worked item in this brief.
• Graphs of |x|/x and x/|x| near 0: two horizontal rays at heights 1 and −1 with hollow ends, plus a filled dot at the origin for the stated value 0.

Where this sits in the book

• NCERT Class XI Mathematics, Chapter 12 "Limits and Derivatives", §12.3 Limits, printed pp. 220–228. The two-clause counterexample and the Summary box are on p. 221; Illustration 9 and Table 12.11 run pp. 226–227.
• Fig 12.3 (p. 221), Fig 12.6 (p. 227).
• Exercise 12.1, items 23, 24, 25, 26, 30, printed pp. 238–239.
• Deliberate cross-references outside this chapter. p. 220 sends the reader to Chapter 2 for two graphs, naming one figure there for x² and another for |x|. The constant function's Chapter 2 figure is named separately, later, inside Illustration 4 on p. 224. All three of those figures are in Chapter 2, not here.
• Chapter Summary, p. 254, which restates the existence criterion.