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Limits are often the first chapter of calculus, and they carry into everything after it: derivatives, continuity, and the area under curves. Many students can do the routine ones and then freeze when a question looks unfamiliar. This post gives you a short decision process, the standard limits worth knowing by heart, worked examples, and the mistakes that cost marks.
What do JEE Main and JEE Advanced include?
The NTA Syllabus for JEE (Main) 2026 puts "Limits, continuity and differentiability" in Unit 7. The JEE (Advanced) 2026 Information Brochure is more specific: limit of a function at a real number, continuity, limits of sums, products and quotients, "L'Hospital rule of evaluation of limits of functions", continuity of composite functions and the intermediate value property. In NCERT, limits are in Class 11, Chapter 12 and continuity in Class 12, Chapter 5. Check the current year's syllabus before you plan.
What are the indeterminate forms?
Substitute first. If you get a number, you are done. If you get one of these, you have work to do:
0/0 and ∞/∞
∞ − ∞ and 0 × ∞
1^∞, 0⁰ and ∞⁰
"Indeterminate" means the form alone does not tell you the answer. Two different 0/0 limits can give 3, 0 or no limit at all.
Which standard limits should you know?
As x → 0:
sin x / x → 1 and tan x / x → 1 (x in radians)
(1 − cos x) / x² → 1/2
(eˣ − 1) / x → 1 and (aˣ − 1) / x → ln a
ln(1 + x) / x → 1
(1 + x)^(1/x) → e
And as x → a: (xⁿ − aⁿ) / (x − a) → n aⁿ⁻¹.
Worked examples
1. A 0/0 trigonometric limit. Find the limit of (1 − cos 4x) / x² as x → 0.
Rewrite it as 16 × (1 − cos 4x) / (4x)². The fraction is the standard form with 4x in place of x, so it tends to 1/2. The answer is 16 × 1/2 = 8.
2. A polynomial 0/0. Find the limit of (x³ − 8) / (x − 2) as x → 2. This is (xⁿ − aⁿ)/(x − a) with n = 3, a = 2, so the answer is 3 × 2² = 12. Factorising x³ − 8 = (x − 2)(x² + 2x + 4) gives the same 12.
3. An ∞ − ∞ form. Find the limit of √(x² + x) − x as x → ∞. Multiply and divide by √(x² + x) + x:
x / (√(x² + x) + x) = 1 / (√(1 + 1/x) + 1) → 1/2.
4. A 1^∞ form. Find the limit of (1 + 3x)^(2/x) as x → 0. When f → 1 and g → ∞, the limit of f^g is e raised to the limit of g(f − 1). Here g(f − 1) = (2/x)(3x) = 6, so the answer is e⁶.
5. L'Hospital's rule (named in the JEE Advanced syllabus). Find the limit of (eˣ − 1 − x) / x² as x → 0. It is 0/0, so differentiate top and bottom: (eˣ − 1) / 2x, still 0/0. Again: eˣ / 2 → 1/2.
How do you check continuity at a point?
A function f is continuous at x = a when the limit of f(x) as x → a exists and equals f(a). In practice, check three numbers: the left-hand limit, the right-hand limit and f(a).
Example. f(x) = ax + 3 for x ≤ 1 and 2x² − a for x > 1. Find a so that f is continuous.
Left limit and f(1): a + 3.
Right limit: 2 − a.
Set a + 3 = 2 − a, so a = −1/2. Check: both sides give 2.5.
Why does the graph picture mislead?
In a classic paper, Tall and Vinner (1981) separated a student's "concept image", everything they associate with an idea, from the formal "concept definition". They asked 41 university entrants with A or B grades in A-level mathematics which of five functions were continuous. For f(x) = 1/x (x ≠ 0), 35 said discontinuous, with reasons like "the graph is not in one piece". Yet by the definition, 1/x is continuous at every point of its domain, and NCERT's Class 12 chapter works this out as Example 9. A related study, Williams (1991), found that students' picture of a limit as something "approaching" was "extremely resistant to change".
For JEE, the lesson is concrete: answer continuity questions from the definition, at points in the domain, not from whether you can draw the graph without lifting your pen.
Common mistakes and how to avoid them
Treating 1^∞ as 1. It is indeterminate. Use the e^(g(f − 1)) rule.
Degrees in sin x / x. The standard limit holds only in radians.
Treating ∞ − ∞ as 0. Rationalise or take out the highest power first.
Using L'Hospital's rule on a form that is not 0/0 or ∞/∞. Check the form each time you differentiate.
Forgetting one side. For piecewise functions and |x|, always compute both one-sided limits.
When a limit has you stuck, it helps to see each step justified. The Fermi Problem Library has a step-by-step solver that explains the reasoning rather than printing an answer, so you can see which standard limit a rewrite was aiming for. If the functions themselves are shaky, start with our post on domain and range of functions.
A practice routine for this chapter
Week 1: Write the standard limits from memory each day until you get all of them right twice.
Week 1: Twenty mixed limits. Before solving, write only the indeterminate form for each.
Week 2: Ten continuity and differentiability questions on piecewise functions. Compute both one-sided limits every time.
Week 2: One timed set of past JEE questions on limits, with every error labelled as wrong form, wrong standard limit or algebra.
Tomorrow, take five limits from your textbook and, before solving any, write the indeterminate form next to each.
