How to Study Limits, Continuity and Differentiability for JEE | Complete PYQ Analysis (JEE Main 2024–2026 & JEE Advanced 2015–2026)

If you are wondering how to study Limits, Continuity and Differentiability for JEE, start with two numbers. JEE Main asked 75 questions from this chapter across 2024, 2025 and 2026, and JEE Advanced asked 39 questions from it between 2015 and 2026, with at least one in every single year. This guide breaks those questions down by sub-topic, based on the papers conducted by the National Testing Agency (jeemain.nta.ac.in) for JEE Main and the JEE Advanced papers from 2015 to 2026.

We went through every one of those questions and sorted each into one of five groups: evaluating limits, continuity at a point, counting points of discontinuity and non-differentiability, differentiability, and properties of continuous and differentiable functions. The tables below show exactly where the marks come from, and the rest of the guide shows how to win them.

This chapter is also the doorway to all of Calculus. Derivatives, Application of Derivatives, Definite Integration and Differential Equations all assume you can evaluate a limit and test continuity without hesitating. Weakness here shows up as lost marks in four other chapters.

1. Why Limits, Continuity and Differentiability Is High Priority for JEE

It gives you one or two questions in almost every JEE Main shift. JEE Main had 28 questions from this chapter in 2024, 26 in 2025 and 21 in 2026, between 4.42% and 5.47% of all Maths questions in each year. That is roughly 1.1 to 1.4 questions per paper, every paper.

It is the most consistent chapter in JEE Advanced. Over 2015–2026, JEE Advanced asked 39 questions from this chapter and never skipped a year. In 2025 it carried 12.5% of the Maths paper and in 2024 it carried 11.76%. Very few chapters can claim twelve years in a row.

The marks are technique marks. Most JEE Main questions here are not conceptually deep. They test whether you can expand a function in a series, handle a 1^∞ form, match left- and right-hand limits, or count corner points of a modulus function quickly and without slips. Technique can be drilled, which makes this one of the most reliable chapters to improve in a few weeks.

"Students think limits are about tricks. They are not. Ninety percent of JEE limits fall to three tools: standard limits, series expansion and the 1^∞ formula. Learn those three properly and you stop guessing which trick to use." — AKS Yadav Sir, Maths HOD (IIT BHU, AIR 1 FTRE 2009, Ex-Faculty Allen Delhi)

2. JEE Main PYQ Analysis 2024–2026

Any plan for how to study Limits, Continuity and Differentiability for JEE should start from the papers. Here is how many questions JEE Main asked from this chapter in each of the last three years.

And here are the same 75 questions sorted by what they actually test.

*The text of five questions in the public archive did not render fully, so we could not classify them with confidence. They are counted in the yearly totals only. Questions that test two ideas are counted once, under the idea they mainly test.

Evaluating limits is the single biggest block: 28 of 75 questions, more than a third of the chapter. Continuity and counting problems together give another 29. In other words, about four in five questions in this chapter come from three skills: evaluating a limit, matching left- and right-hand limits, and spotting where a function breaks or has a corner.

5 key patterns from recent JEE Main papers

Pattern 1: Series expansion is now the default tool for limits. 2025 asked for the limit of (tan⁻¹x + ½ ln((1+x)/(1−x)) − 2x)/x⁵, which cannot be done sensibly with L'Hôpital's rule. Expanding gives 2x + (2/5)x⁵ + … in the numerator, so the limit is 2/5. If your expansions of sin, tan, tan⁻¹, e^x and ln(1+x) are not automatic, these questions eat your time.

Pattern 2: "Find the parameter" limits. Many questions give a limit that must be finite or must equal a value, and ask for the constants. 2025 had "if the limit is finite, find …" and "for α, β, γ, if the limit equals …", and 2026 had "the product of all possible values of α for which the limit equals …". The method is always the same: expand, make the lower powers vanish, then match the first surviving term.

Pattern 3: Greatest integer and fractional part functions everywhere. Across 2024–2026, [x] or {x} appeared in continuity questions, counting questions and even limit questions (for example, left- and right-hand limits of a fractional-part function at a point in 2024, and a least-natural-number limit with [t] in 2025). Learn exactly how [x] behaves just to the left and right of an integer.

Pattern 4: Counting questions are a favourite numerical. Every year has asked for the number of points where a function is discontinuous or not differentiable, often as "m + n" or a similar combination. 2024 asked it for f(x) = |2x² + 5|x| − 3|, 2025 with greatest-integer functions, and 2026 with a max function and with [x] on an interval. These are fast marks once you can sketch the function.

Pattern 5: Limits hide inside other chapters. Limits of sums that become series (2025, Σ_r=1ⁿ terms as n → ∞), limits involving the roots of a quadratic (2024 and 2025), and a function defined as a limit of a telescoping tan series (2025) all sit in this chapter. Expect a limit wrapped around Sequences and Series or Quadratic Equations.

3. JEE Advanced PYQ Pattern 2015–2026

This is the 12-year view that most guides skip. JEE Advanced has asked this chapter every year from 2015 to 2026.

*Five questions whose archive text did not render fully are counted in the totals but not classified.

For Advanced aspirants, this table changes how to study Limits, Continuity and Differentiability for JEE. Main is dominated by evaluating limits, but Advanced gives almost as much weight to differentiability (10 questions) as to limits (11). Advanced also loves multi-correct statement questions: 18 of the 39 were "one or more correct" questions, where you must judge every option on its own.

4. Complete Sub-topic Breakdown

The smartest way to plan this chapter is to match your time to each sub-topic, because they are very different in size and difficulty. Use the time column in your mock tests: if you are consistently slower, that sub-topic needs more practice, not more reading.

Difficulty and target time are our faculty's guidance for exam practice, not exam data.

Evaluating limits covers standard limits, series expansions, the 1^∞ form, limits at infinity, limits of sums and "find the constants" questions. It is the backbone of the chapter in JEE Main.

Continuity at a point is almost always the same question: a piecewise function, a parameter, and the condition LHL = RHL = f(a). The hard part is usually the limit inside it.

Counting discontinuity and non-differentiability needs you to sketch: modulus functions, max/min functions and greatest-integer functions. Every corner and every jump is a point to count.

Properties of continuous and differentiable functions covers functional equations, twice-differentiable functions with given conditions, and mean-value or intermediate-value arguments. These are the hardest questions in the chapter and the most common in Advanced.

Differentiability asks whether LHD equals RHD, or which parameter makes a function differentiable. Advanced pushes this further with products like x·f(x) and composites.

5. Common Mistakes in Limits, Continuity and Differentiability

Good preparation also means knowing what not to do. These are the mistakes we see most often when students solve this chapter's PYQs.

6. Important Formulas by Sub-topic

Keep this table open while you practise. Most of the chapter runs on these results, which is why success here is mostly about applying them quickly and without errors. Make your own copy by hand: writing the expansions out is the fastest way to remember them.

7. How to Study Limits, Continuity and Differentiability for JEE — Mastering Each Sub-Topic

This is the core of how to study Limits, Continuity and Differentiability for JEE: one sub-topic at a time, starting with the ones that have given the most marks. Do not move on until you can solve the PYQ type listed under each heading without notes.

Evaluating limits (28 Main Qs 2024–26 · 11 Advanced Qs)

Before you solve any limit, substitute and name the form: 0/0, ∞/∞, ∞ − ∞, 0·∞, 1^∞, 0⁰ or ∞⁰. The form decides the tool.

• 0/0 with trigonometric, exponential or log terms: use the standard limits if the expression is simple; otherwise expand in series. Series expansion is the safest method for JEE because it never loops, unlike repeated L'Hôpital.
• ∞/∞ at infinity: divide by the highest power. For exponentials and powers together, take out the dominant factor, as in the 2025 question with (3x − 1)^x/2 and √((3x + 2)^x).
• 1^∞: use e^{lim g(f − 1)} straight away.
• ∞ − ∞ with roots: rationalise. The 2025 question with cosec x · (√(2cos²x + 3cos x) − √(cos²x + sin x + 4)) is exactly this.
• Limits of sums: write the general term as (1/n) f(r/n) and convert to a definite integral, or telescope.

For "find the parameter" questions, expand everything to the power of the denominator, set the coefficients of lower powers to zero, and match the first non-zero coefficient to the given value. JEE Advanced 2025 used exactly this with an integral: ∫₀^x dt/(1 − t²) = ½ ln((1 + x)/(1 − x)) = x + x³/3 + …, combined with βx cos x and divided by x³.

Practice target: write the eight expansions in the formula table from memory in under two minutes, then solve 20 PYQ limits using expansion only.

"When a student is stuck on a limit, I ask one question: what is the power of x in the denominator? Expand everything up to that power and the answer falls out. It works on the hardest Advanced limits as well as the easiest Main ones." — AKS Yadav Sir, Maths HOD (IIT BHU, AIR 1 FTRE 2009, Ex-Faculty Allen Delhi)

Continuity at a point (15 Main Qs 2024–26 · 1 Advanced Q)

Almost every Main question here looks like this: f(x) is defined by one expression for x ≠ a (or x < a and x > a) and a constant at x = a; find the constant (or a combination of constants) that makes f continuous. The method:

• Find the left-hand limit and the right-hand limit separately. If the formula is the same on both sides, one limit is enough.
• Set LHL = RHL = f(a) and solve.
• If [x] or {x} appears, replace it with its actual value on each side before taking the limit. This is where the 2024 ordered-pairs question and the 2026 greatest-integer questions catch students.

The hard part of a continuity question is usually the limit inside it, so this sub-topic improves automatically as your limit skills improve. For the intermediate value property (asked once in Advanced, 2017): if f is continuous on [a, b] and f(a), f(b) have opposite signs, f has a root in (a, b). Apply it by building a new function g(x) and checking its sign at the end points.

Practice target: 15 piecewise continuity PYQs, each solved by writing LHL, RHL and f(a) as three separate lines.

Counting points of discontinuity and non-differentiability (14 Main Qs 2024–26 · 7 Advanced Qs)

This sub-topic is about drawing, not algebra. The checklist:

• [x]: jumps at every integer. [g(x)] jumps wherever g(x) crosses an integer value, unless g has a turning point there that only touches the integer.
• |g(x)|: corners where g changes sign with non-zero slope.
• f(|x|): reflect the right half; check x = 0 for a corner.
• max{f, g} and min{f, g}: corners where the graphs cross with different slopes.
• Combinations: where a [x] jump is multiplied by a factor that is zero at that point, the jump can disappear. Always check.

Worked example from JEE Main 2024: f(x) = |2x² + 5|x| − 3|. Put t = |x|: 2t² + 5t − 3 = (2t − 1)(t + 3), which is zero at t = ½, so x = ±½. At these points the inner function changes sign, giving two corners. At x = 0, the inner function equals −3, so |inner| = 3 − 5|x| − 2x² near 0, which has a corner at 0 because of |x|. So f is continuous everywhere and not differentiable at 3 points.

Practice target: sketch 15 functions from PYQs and mark every jump and corner before reading the options.

"Counting questions punish students who skip the sketch. A rough graph takes 40 seconds and shows every corner. Algebra alone misses the point at x = 0 every time." — AKS Yadav Sir, Maths HOD (IIT BHU, AIR 1 FTRE 2009, Ex-Faculty Allen Delhi)

Properties of continuous and differentiable functions (7 Main Qs 2024–26 · 5 Advanced Qs)

These are the questions that give you an unknown function and some conditions: "f is twice differentiable with f(0) = …, f'(0) = …", "f is continuous and satisfies f(x + y) = …", or "f is differentiable on [a, b] with …; find c such that …". Learn three toolkits:

• Functional equations: put x = y = 0 first, then y = x, then differentiate with respect to one variable. For f(x + y) = f(x) + f(y) + f(x)f(y) (JEE Advanced 2020), notice that 1 + f behaves like an exponential.
• Derivative from first principles: f'(x) = lim_h→0 [f(x + h) − f(x)]/h. When a question gives f(x + y) in terms of f(x) and f(y), build f'(x) this way.
• Rolle, LMVT and IVT: these turn a condition on f into the existence of a point c. Questions in 2024 and Advanced 2017–2018 use exactly this reasoning.

Practice target: 10 functional-equation PYQs and 10 "twice differentiable f" PYQs. Write the key substitution at the top of each before you start.

Differentiability (6 Main Qs 2024–26 · 10 Advanced Qs)

JEE Main asks this less often, but JEE Advanced asks it more than anything except limits. The core skills:

• Test LHD and RHD at the joining point of a piecewise function. Differentiable implies continuous, so check continuity first; if it fails, you are done.
• Products with |x − a|: |x − a|·g(x) is differentiable at a exactly when g(a) = 0. 2025 used this idea with (x² − 1)|x² − ax + 2| being non-differentiable at exactly two points.
• Oscillating functions: xⁿ sin(1/x) (with value 0 at x = 0) is continuous at 0 for n ≥ 1 and differentiable at 0 for n ≥ 2. Advanced 2025 asked about the differentiability, monotonicity and local minimum at 0 of a function with an oscillating term, so know these results cold.
• g(x) = x f(x): Advanced 2026 asked what continuity of f at 0 implies for g, and the reverse. g is always continuous at 0 if f is bounded near 0; if f is continuous at 0, g is differentiable at 0 with g'(0) = f(0).

Practice target: every JEE Advanced differentiability question from 2015 to 2026. There are only ten, and they show every trick Advanced uses.

"In Advanced, differentiability questions are really questions about definitions. Write the definition of the derivative at the point, then simplify. Students who try to differentiate the formula directly at a joining point lose these marks." — AKS Yadav Sir, Maths HOD (IIT BHU, AIR 1 FTRE 2009, Ex-Faculty Allen Delhi)

That is how to study Limits, Continuity and Differentiability for JEE sub-topic by sub-topic. Once you have finished all five, test yourself under exam conditions with our JEE test series, and bring any question that still does not make sense to the JEE Doubt Lab for a 1-on-1 explanation.

8. JEE Advanced Exclusive Topics

If you are deciding how to study Limits, Continuity and Differentiability for JEE Advanced as well as Main, this section is for you. Advanced does not have a separate syllabus for this chapter, but it goes deeper in five places.

Multi-correct statement questions

18 of the 39 Advanced questions from 2015 to 2026 were "one or more correct". Each option is a separate true/false judgement about continuity, differentiability or a limit. Practise by writing a one-line proof or a counter-example for every option, even the ones you think are obvious.

Implications between continuity and differentiability

Advanced 2026 asked, for g(x) = x f(x) with f arbitrary, whether continuity of f at 0 makes g differentiable, and whether differentiability of g forces f to be continuous. You need counter-examples ready: f(x) = sign-type functions, f(x) = sin(1/x), and |x|.

Functions defined through limits

Advanced 2019 defined PROPERTY 1 and PROPERTY 2 through limits, Advanced 2022 asked for the limit of f_n, and Main 2026 defined f(x) itself as a limit. Treat the limit as the function's formula, simplify it first, and only then test continuity or differentiability.

Match-the-list on one function

Advanced 2018, 2024 and 2025 each had a match-the-list question that asked several things about one function: a continuity parameter, monotonicity, local extrema and the number of non-differentiable points. These mix this chapter with Application of Derivatives, so finish both chapters before you attempt them.

Abstract twice-differentiable functions

Advanced 2017 and 2018 gave functions only through conditions like f'' > 0, or f' = e^{f − g}·g'. Convexity arguments, Rolle's theorem on a cleverly built auxiliary function, and integrating a derivative relation are the standard tools.

9. Revision Checklist

The last step in how to study Limits, Continuity and Differentiability for JEE is an honest self-check. You should be able to answer "yes" to every line below without notes.

• ☐ I can name the indeterminate form of any limit before I start solving it.
• ☐ I can write the series expansions of sin x, cos x, tan x, e^x, ln(1 + x), tan⁻¹x, sin⁻¹x and (1 + x)ⁿ from memory.
• ☐ I can solve any 1^∞ limit with the e^{lim g(f − 1)} formula.
• ☐ I can find constants from a "the limit is finite / equals k" condition.
• ☐ I can convert a limit of a sum into a definite integral.
• ☐ I can write the left and right values of [x] and {x} at any integer.
• ☐ I can check continuity using LHL, RHL and f(a) separately.
• ☐ I can count corners of |f(x)|, f(|x|), max and min functions by sketching.
• ☐ I can test differentiability at a joining point with LHD and RHD.
• ☐ I know when |x − a|·g(x) is differentiable at a.
• ☐ I can solve the standard functional equations and use Rolle, LMVT and IVT.
• ☐ I have solved every JEE Main 2024–2026 and JEE Advanced 2015–2026 question from this chapter at least once.

Conclusion

Limits, Continuity and Differentiability gave JEE Main 75 questions from 2024 to 2026 and appeared in every JEE Advanced exam from 2015 to 2026, 39 questions in all. Evaluating limits alone gave 28 of the Main questions, and continuity plus counting problems gave another 29. So if you are working out how to study Limits, Continuity and Differentiability for JEE, the order is clear: master standard limits and series expansions first, then continuity at a point, then counting corners and jumps by sketching, and finally the differentiability and abstract-function questions that Advanced loves.

How we built this data. We went through every JEE Main shift paper from 2024, 2025 and 2026 and every JEE Advanced paper from 2015 to 2026, picked out each question from this chapter and tagged it by the main idea it tests. Where the archived text of a question was too incomplete to classify with confidence, we counted it in the totals and said so instead of guessing. No number on this page is estimated.

If you want a faculty member to check your preparation, every pack starts with a free 30-minute mentorship call. See all Maths chapter packs on our JEE Maths coaching page, and once your exam is done, our JEE counselling team can help you turn your rank into the right college and branch.

Frequently Asked Questions

Related Chapter Guides