JEE Main Maths · Limits, Continuity & Differentiability PYQ

JEE Main Limits, Continuity & Differentiability PYQ (2002–2025)

Q: How many Limits PYQs should I solve?

Target 60–80 PYQs across 2010–2025. At ~7% weightage, Limits is a high-ROI chapter. The standard-limit patterns are highly repetitive — volume builds instant recall.

Q: When should I use L'Hôpital vs Taylor series?

L'Hôpital: for straightforward 0/0 or ∞/∞ where derivatives are easy. Taylor series: for limits involving sin, cos, ln, eˣ near 0 — more elegant and often faster.

Q: Are Limits PYQs harder than Differentiation PYQs?

Comparable difficulty but different flavours. Limits are more pattern-recognition. Differentiation has more mechanical chain-rule application with maxima-minima twists.

Q: What's the most-tested Limits PYQ pattern?

Trigonometric limits near 0 with small-angle approximations — lim (1-cos ax)/(1-cos bx) type. Roughly 4–5 variants per 10-year window.

Q: How do I verify continuity quickly in PYQs?

For piecewise function at x = a: compute f(a), compute LHL (from x = a-h, h→0), compute RHL (from x = a+h, h→0). All three equal → continuous. Any mismatch → discontinuous.

Q: Does Limits connect to Integration PYQs?

Yes — definite integrals as limits of sums (Riemann sum interpretation) appears in a few PYQs per 10-year window. Foundation for understanding Integration rigorously.

Limits, Continuity & Differentiability is a P1 Calculus-foundation chapter — ~7% JEE Main weightage with 2 questions per paper. It's the conceptual gateway to Differentiation and Integration — strong fluency here cascades into the rest of Calculus.

Limits, Continuity & Differentiability PYQs from 2002 to 2025 covering evaluation techniques (L'Hôpital, standard limits, Taylor expansion), continuity at a point, differentiability, and problems combining the three concepts. Every KaTeX-rendered solution names the technique upfront.

Practise Limits, Continuity & Differentiability PYQs All Maths PYQs

Limits, Continuity & Differentiability at a Glance

Weightage

~7%

approx · 10-yr avg

Priority

Must master

Year range

2002–2025

PYQ coverage

Typical in paper

per session

Key Sub-Topics & What's Tested

Standard Limits

lim(x→0) sin x / x = 1, lim(x→0) (1 - cos x)/x² = 1/2, lim(x→∞) (1 + 1/x)^x = e, lim(x→0) (aˣ - 1)/x = ln a.

Evaluation Techniques

Direct substitution, factoring, rationalising, L'Hôpital's rule for 0/0 and ∞/∞, Taylor series expansion.

L'Hôpital's Rule

For 0/0 or ∞/∞ forms, lim f(x)/g(x) = lim f'(x)/g'(x). Apply iteratively until indeterminate form resolves.

Trigonometric Limits

Limits involving sin, cos, tan near 0; using sin x ≈ x, cos x ≈ 1 - x²/2 for small angles.

Exponential & Logarithmic Limits

Limits using eˣ = 1 + x + x²/2 + ..., ln(1+x) = x - x²/2 + ..., for small x.

Continuity at a Point

Three conditions: f(a) defined, lim(x→a) f(x) exists, both equal. LHL = RHL = f(a).

Differentiability at a Point

LHD = RHD at x = a. Differentiability implies continuity but not vice versa.

Piecewise Functions

Checking continuity and differentiability at boundary points of piecewise-defined functions — very common PYQ template.

Question Type Distribution

Question Type	Share (approx)	Example Pattern
Limit Evaluation (Algebraic)	20%	Evaluate lim(x→2) (x² - 4)/(x - 2) using factoring.
L'Hôpital Application	25%	Evaluate lim(x→0) (sin x - x)/x³ using L'Hôpital twice.
Trigonometric Limits	20%	Evaluate lim(x→0) (1 - cos 2x)/(x sin x).
Exponential/Logarithmic Limits	10%	Evaluate lim(x→∞) (1 + 2/x)^(3x).
Continuity Verification	15%	For f(x) = {x² if x<1; ax + b if x≥1}, find a and b such that f is differentiable at x = 1.
Differentiability at a Point	10%	Check if f(x) = \|x³\| is differentiable at x = 0.

How to Solve Limits, Continuity & Differentiability PYQs

1
Memorise 10 standard limits. sin x/x → 1, (1-cos x)/x² → 1/2, tan x/x → 1, (eˣ-1)/x → 1, ln(1+x)/x → 1, (1+1/x)^x → e, (1+x)^(1/x) → e. Direct recall saves 90 seconds per question.
2
Check indeterminate form first. If direct substitution gives 0/0, ∞/∞, 0·∞, ∞-∞, use L'Hôpital or algebraic manipulation. Otherwise, direct substitution works.
3
For piecewise functions, use LHL and RHL. Evaluate f(a-h) and f(a+h) separately, equate for continuity. For differentiability, evaluate LHD and RHD.
4
Taylor series for hard limits. Expand sin x = x - x³/6 + ..., cos x = 1 - x²/2 + ..., eˣ = 1 + x + x²/2 + .... Keep terms up to needed order.
5
Differentiability implies continuity, not vice versa. |x| is continuous at 0 but not differentiable. This concept drives many trick PYQs.

Common Mistakes That Cost Marks

Applying L'Hôpital to non-indeterminate forms. lim(x→0) (sin x)/x² is 0/0 at first glance, but becomes cos(x)/2x after L'Hôpital — still needs care. Always verify form.
Forgetting to check both LHL and RHL. A limit exists only if LHL = RHL = a finite number. If they differ, limit does not exist.
Confusing f(a) with lim f(x) at x = a. f(a) is the function value. lim f(x) is the limit. Continuity requires both to exist and be equal.
Using Taylor series without enough terms. For lim (sin x - x)/x³, you need sin x = x - x³/6 + O(x⁵). Using only x - x³/6 gives answer -1/6; using just sin x ≈ x gives 0 (wrong).
Assuming differentiable → continuous works both ways. Continuous does NOT imply differentiable. Function must have no sharp corners or cusps for differentiability.

JEE Main Maths PYQ Chapter-Wise

Browse PYQs across all Maths chapters.

JEE Main Differentiation PYQ

Next Calculus chapter — uses continuity and differentiability foundations.

JEE Main Integration PYQ

Top-weightage Calculus chapter — connects via Riemann sum limit.

Top High-Weightage Maths Chapters (data-backed)

Where Limits sits in the Calculus-first priority order.

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Frequently asked questions

How many Limits PYQs should I solve?