Math notesCalculusLimits
Independent topic page
Full marks here mean: one-sided limits table → continuity vs differentiability as separate verdicts → simplify before L’Hôpital → never apply L’Hôpital without 0/0 or ∞/∞.
01. Limit exists iff LHL=RHL. Continuity needs limit = f(a). Differentiability needs LHD=RHD (finite). Differentiable ⇒ continuous; converse false (|x| at 0).
02. Standard limits (sinx/x, etc.) and algebraic cancellation beat illegal L’Hôpital. State the indeterminate form explicitly when you use L’Hôpital.
03. Rolle/MVT: check hypotheses (continuity on closed, differentiability on open, f(a)=f(b) for Rolle) before concluding existence of c.
04. Common process on Advanced: list break points → fill table → conclude → only then expansions/L’Hôpital if still indeterminate.
05. Advanced lane: continuity vs differentiability at a point is a left/right table — f, f', limits side by side. Philosophy language loses marks; bookkeeping wins.
06. Indeterminate forms: stamp 0/0 or ∞/∞ before L’Hôpital. Algebraic simplify (factor, rationalise, standard trig limits) first; L’Hôpital only when still indeterminate.
Official · 2026
Official syllabus points for Limits · Continuity · Differentiability only — not the full Calculus portion map. Portion hub: Calculus · Calculus analysis.
No topic-specific exclusions flagged — finish the unit points below for Limits.
Source: https://jeeadv.ac.in/documents/jee-advanced-2026-syllabus.pdf
Exam focusLimits, continuity/differentiability checks, and L’Hospital — not the full AOD unit.
Syllabus points
Do / check
2020–2026 paper weight
Topic match inside parent portion Calculus (28.5% of Math tags). Stats below are portion backdrop; matched subtopics show where Limits · Continuity · Differentiability sits.
Portion-wide type/difficulty mix lives on the Calculus analysis — not repeated here so this page stays on Limits.
01. Dominant subtopics: AOD (18), Definite Integrals (11), Continuity/Differentiability (9), Functions (7).
Cross-topic stems
Advanced rarely asks a chapter in isolation. These are tagged multi-idea Math stems that touch Limits · Continuity · Differentiability — practise the knowledge bridge, not only the single-chapter drill.
Step 1: analyse the given limit expression to identify the correct f(x) / derivative behaviour. Step 2: on a related (or following) stem, sketch the region defined by inequalities. Step 3: integrate carefully (logs appear from 1/x-type boundaries).
Back-to-back calculus depth: analytic limit then geometric area.
Pair every limit/AOD drill with one area-of-region numerical the same day.
Step 1: for the ellipse–parabola common-tangent MC, write both tangent conditions and compare. Step 2: later area NUM — find intersection limits first. Step 3: integrate difference of functions; keep trig-equation NUM technique nearby for neighbouring items.
Classic Advanced join: geometry conditions feeding calculus numerics in the same paper.
Any week you revise conics, end with one area-between-curves NUM.
Step 1: for the area item, find intersection limits and integrate. Step 2: for the parabola item, apply focus–directrix or tangent properties. Step 3: notice how often Advanced seats Calculus and Conics as neighbours — train that adjacency.
2020’s Calculus weight makes area items feel easy until limits are wrong.
Always find limits from geometry before writing ∫.
Why methods matter
This chapter decides whether you trust your Calculus. These methods force the LHL/RHL table and ban illegal L’Hôpital — exactly what Advanced tests.
Makes continuity vs differentiability feel like bookkeeping, not philosophy.
Do thisLHL/RHL/f(a)/LHD/RHD in a tiny table at each break.
Matches how Advanced grades continuity vs differentiability separately.
At a piecewise join, fill five cells before talking about ‘smoothness’.
Protects you from illegal L’Hôpital — Advanced designs those traps on purpose.
Do thisCancel factors / use conjugates / standard limits first.
Faster and safer than differentiating messy quotients.
(√(1+x)−1)/x → conjugate → 1/(√(1+x)+1) → 1/2 as x→0.
Rolle/MVT MC becomes a tick list — conclusions without hypotheses get rejected.
Do thisFor Rolle/MVT MC, tick each hypothesis true/false.
Many options fail only on one missing hypothesis.
If not differentiable on (a,b), Rolle conclusion is not guaranteed.
Forces you to say 0/0 or ∞/∞ out loud before differentiating a quotient.
Do thisWrite ‘0/0’ or ‘∞/∞’ before L’Hôpital.
Prevents illegal applications that Advanced traps.
If limit is 2/5 already, do not differentiate.
Processed examples
Read the impact line first. Then notice which family/model was chosen. The solution is written the way your rough sheet should look in the hall.
Evaluate lim x→0 (√(1+x)−1)/x.
Illegal L’Hôpital is a trap; conjugate simplification is the bank move.
Model chosenMultiply conjugate before any derivative.
Answer1/2
Takeaway: Simplify first; L’Hôpital only after stamping 0/0.
f(x)=|x| at x=0: continuous? differentiable?
Continuity and differentiability are separate verdicts — Advanced grades both.
Model chosenTable LHL/RHL/f; then LHD/RHD.
AnswerContinuous, not differentiable
Takeaway: Never claim differentiability at a corner.
f(x) = sin x / x (x < 0), f(0) = a, f(x) = (e^{bx} − 1)/x (x > 0). Can any real a, b make f differentiable at 0?
Continuity fixes parameters; differentiability can still fail — Advanced loves that two-gate pattern.
Model chosenMatch LHL/RHL/f first; then compare LHD and RHD.
AnswerNo — no real a, b makes f differentiable at 0
Takeaway: After matching values, compare first-order one-sided derivatives.
Reusable models
Advanced stems cluster. Name the family in 10–15s, write the fast model, then run the same path. Do not invent a new plan each time.
Looks likef defined differently on intervals; ask at join / floor / |x|
Fast modelTiny table: LHL, RHL, f(a), then LHD, RHD — separate verdicts.
Table replaces vague ‘looks smooth’ arguments.
Do not apply a single smooth formula across a kink.
Looks likeQuotient or difference that vanishes at the point
Fast modelSimplify / rationalise / standard limits FIRST; L’Hôpital only after stamping 0/0 or ∞/∞.
Algebraic cancel often finishes before any derivative.
Do not L’Hôpital a limit that is already determinate.
Looks likeWhich theorem applies / existence of c
Fast modelTick hypotheses one-by-one; conclusion only if all hold.
Hypothesis checklist is the whole question.
Do not quote the conclusion when a hypothesis fails.
Looks likelim (1+u)^{1/u}; lim (a^x−1)/x; forms that become 1^∞ after rewrite
Fast modelRewrite to exp(lim ln) or standard (e^x−1)/x → 1; stamp the form first.
Form stamp chooses exp/ln vs L’Hôpital in 10s.
Do not L’Hôpital a 1^∞ before taking ln.
Hall timing
Floor integers, | | kink, piecewise joins are the usual suspects.
For limits: simplify algebraically/trig first. Only claim 0/0 or ∞/∞ if truly indeterminate.
MC: each statement gets its own T/F line (continuous? differentiable?).
If MVT/Rolle, tick hypotheses explicitly before using the conclusion.
After the model is locked
lim x→a f = L iff LHL = RHL = L
Two-sided limits
f(a) defined, lim exists, lim = f(a)
Continuity checks
f′(a)=lim h→0 [f(a+h)−f(a)]/h
First principles / one-sided derivatives
lim θ→0 sinθ/θ = 1 (radians)
Trig limits
If 0/0 or ∞/∞, lim f/g = lim f′/g′ (when latter exists)
Indeterminate quotients only
f cont [a,b], diff (a,b), f(a)=f(b) ⇒ ∃c f′(c)=0
Existence of horizontal tangent
f cont [a,b], diff (a,b) ⇒ ∃c f′(c)=[f(b)−f(a)]/(b−a)
Mean slope realisation
g≤f≤h and lim g=lim h=L ⇒ lim f=L
Oscillatory / boundable limits
Make it automatic