NDA Maths · Teaching 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.
Subtopic notes
Evaluating Limits — Factor, Rationalise, Standard Forms
31 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.
Open note
One-Sided, Greatest-Integer & Modulus Limits
16 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.
Open note
Continuity & Differentiability
34 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.
Open note
PYQ weightage by concept
12 concepts · 81 PYQs — where the marks actually sit, so you know what to drill first
PYQ weightage by concept
12 concepts · 81 PYQs — where the marks actually sit, so you know what to drill first
| Concept | PYQs | Share |
|---|---|---|
| 0/0 by factoring, cancelling, and rationalising | 13 | 16% |
| The standard limits to memorise | 9 | 11% |
| What a limit is — and when it exists | 4 | 5% |
| L'Hôpital's rule for 0/0 and ∞/∞ | 3 | 4% |
| The 1^∞ form | 2 | 2% |
| Concept | PYQs | Share |
|---|---|---|
| Left-hand and right-hand limits | 7 | 9% |
| Limits of the greatest-integer function | 7 | 9% |
| Limits involving the modulus | 2 | 2% |
| Concept | PYQs | Share |
|---|---|---|
| Finding parameters so f is continuous | 11 | 14% |
| Continuity vs differentiability | 11 | 14% |
| The definition of continuity | 7 | 9% |
| Types of discontinuity (removable, jump, oscillatory) | 5 | 6% |
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 memorise7 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 |
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)