NDA Maths · Free notes
Limits & Continuity — NDA Mathematics
Limits & Continuity is around 81 past-year NDA questions and the gateway to all of calculus — every derivative and integral is a limit underneath. The chapter rewards a small, reliable toolkit: evaluate a 0/0 limit by factoring, rationalising, or a standard form; handle one-sided limits and the greatest-integer / modulus functions where the two sides disagree; and test continuity by checking left limit = right limit = the function's value. Work the three notes in order — first the evaluation techniques, then one-sided and special-function limits, then continuity and its link to differentiability. The traps are predictable: a hidden one-sided mismatch, a greatest-integer jump, or assuming an oscillating function like sin(1/x) has a limit.
The handout has every subtopic, worked example, formula and trap with answers shown. The formula sheet is the chapter’s formulas and traps alone, on one page.
Subtopic notes
Evaluating Limits — Factor, Rationalise, Standard Forms
34 PYQsThe core toolkit for a 0/0 limit: direct substitution first, then factor-and-cancel, rationalise a surd, apply a standard limit, use L'Hôpital, or handle the 1^∞ form via the exponential trick.
One-Sided, Greatest-Integer & Modulus Limits
17 PYQsWhen the function behaves differently on the two sides of a point — a modulus, a greatest-integer step, or a piecewise rule — you must compute the left and right limits separately.
Continuity & Differentiability
36 PYQsA function is continuous at a point when the left limit, the right limit, and the function's value all agree — and continuity is the necessary (not sufficient) condition for differentiability.
Formula & revision sheet
3 formulas · 1 reference tables · 10 gotchas across all subtopics — the exam-eve cheat-sheet
Formula & revision sheet
3 formulas · 1 reference tables · 10 gotchas across all subtopics — the exam-eve cheat-sheet
Formulas (2)
Reference tables (1)
The standard limits to memorise10 rows
| Limit (as x → 0) | Value |
|---|---|
| sin x / x | 1 |
| tan x / x | 1 |
| (1 − cos x) / x² | 1/2 |
| log(1 + x) / x | 1 |
| (eˣ − 1) / x | 1 |
| (aˣ − 1) / x | ln a |
| (1 + x)^(1/x) | e |
| (1 − cos ax) / x² | a²/2 The scale factor SQUARES here — (1 − cos 4x)/x² → 8, not 2. It is the one row where 'sin(ax)/x → a' scaling changes shape. |
| sin ax / sin bx, tan ax / x, (eᵃˣ − 1)/x | a/b, a, a |
| sin x° / x (x in DEGREES) | π/180 Convert first: x° = πx/180 radians. sin x° / tan 3x° → 1/3 after the conversion cancels. |
Watch out for (4)
- sin x / x → 1 only as x → 0→ The standard limits to memorise
- Scale the argument, scale the value→ The standard limits to memorise
- L'Hôpital only on indeterminate forms→ L'Hôpital's rule for 0/0 and ∞/∞
- Differentiate top and bottom separately — not the quotient→ L'Hôpital's rule for 0/0 and ∞/∞
Watch out for (4)
- One-sided limits must AGREE for the limit to exist→ Left-hand and right-hand limits
- ⌊x⌋ jumps at integers — the limit there does NOT exist→ Limits of the greatest-integer function
- x / |x| is +1 on the right, −1 on the left→ Limits involving the modulus
- A square root hides a modulus→ Limits involving the modulus
Watch out for (2)
- Continuity needs the limit to EQUAL the value→ The definition of continuity
- Removable vs jump — the limit's existence is the divider→ Types of discontinuity (removable, jump, oscillatory)
PYQ weightage by concept
12 concepts · 87 PYQs — where the marks actually sit, so you know what to drill first
PYQ weightage by concept
12 concepts · 87 PYQs — where the marks actually sit, so you know what to drill first
| Concept | PYQs | Share |
|---|---|---|
| 0/0 by factoring, cancelling, and rationalising | 14 | 16% |
| The standard limits to memorise | 11 | 13% |
| What a limit is — and when it exists | 4 | 5% |
| L'Hôpital's rule for 0/0 and ∞/∞ | 3 | 3% |
| The 1^∞ form | 2 | 2% |
| Concept | PYQs | Share |
|---|---|---|
| Left-hand and right-hand limits | 7 | 8% |
| Limits of the greatest-integer function | 7 | 8% |
| Limits involving the modulus | 3 | 3% |
| Concept | PYQs | Share |
|---|---|---|
| Continuity vs differentiability | 13 | 15% |
| Finding parameters so f is continuous | 11 | 13% |
| The definition of continuity | 7 | 8% |
| Types of discontinuity (removable, jump, oscillatory) | 5 | 6% |
Test yourself on Limits & Continuity
20 past NDA questions from this chapter, timed at 25 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.