Limits & Continuity for JEE Main: Complete Guide 2026

Limits and Continuity is the gateway to all of JEE Main Calculus, contributing three to five questions every session directly and providing the conceptual foundation for differentiation, integration, and differential equations. The chapter is divided into two almost independent halves: the mechanics of computing limits (standard forms, L'Hôpital, squeeze theorem) and the conceptual analysis of continuity and differentiability at a point. Both halves are heavily tested and require different skills.

Standard Limit Forms You Must Know

JEE expects instant recognition of the following standard limits: lim(x→0) sinx/x = 1; lim(x→0) tanx/x = 1; lim(x→0) (eˣ−1)/x = 1; lim(x→0) ln(1+x)/x = 1; lim(x→∞) (1+1/x)ˣ = e; lim(x→0) (aˣ−1)/x = ln a; lim(x→0) (1+x)^(1/x) = e. These seven results appear, often disguised, in almost every limits question. When a limit produces 0/0 or ∞/∞ form, first try substituting one of these standard limits by algebraic manipulation before reaching for L'Hôpital's rule.

The form 1^∞ is one of the most frequently tested indeterminate forms at JEE. If lim f(x)^g(x) has f→1 and g→∞, rewrite as e^[lim g(x)·(f(x)−1)]. This template resolves every 1^∞ form by reducing it to a standard 0·∞ form, which then becomes 0/0 for L'Hôpital. Practise recognising this 1^∞ template — it eliminates a class of questions that trips many students. Test your limit evaluation speed with a free calculus mock test.

L'Hôpital's Rule and the Squeeze Theorem

L'Hôpital's rule: if lim f(x)/g(x) produces 0/0 or ∞/∞, then the limit equals lim f'(x)/g'(x), provided the latter limit exists. Apply repeatedly if the indeterminate form persists. The rule fails when the resulting limit does not exist but the original limit does — a subtle point that JEE occasionally exploits. Do not apply L'Hôpital to forms other than 0/0 or ∞/∞ without first rewriting.

The squeeze theorem: if g(x) ≤ f(x) ≤ h(x) near x=a, and lim g(x) = lim h(x) = L, then lim f(x) = L. Classic application: lim(x→0) x·sin(1/x) = 0, since −|x| ≤ x·sin(1/x) ≤ |x| and both bounds approach 0. JEE uses squeeze theorem questions to test conceptual understanding rather than calculation — they appear as "which of the following limits can be evaluated by the squeeze theorem" or require applying it to a given function with oscillatory behaviour.

Continuity: Definition and Tests

A function f is continuous at x=a if (1) f(a) is defined, (2) the limit lim_{x→a} f(x) exists, and (3) the limit equals f(a). A function is discontinuous if any condition fails. Types of discontinuity: removable (the limit exists but ≠ f(a) — a "hole" in the graph), jump (left and right limits both exist but are unequal — step functions), infinite (limit → ±∞). JEE asks which type of discontinuity a piecewise function has at a given point, or asks for the value of a parameter that makes a given function continuous.

The key computational tool for piecewise continuity: compute the left-hand limit, right-hand limit, and function value at the junction point. If all three are equal, the function is continuous. For functions involving floor function, fractional part, or absolute value, the junction points are the integers or the zeros of the expression inside. These functions appear in 2-3 limits/continuity questions per session.

Differentiability and Its Failure

A function is differentiable at x=a if and only if it is continuous there AND the left-hand derivative equals the right-hand derivative. The left-hand derivative LHD = lim_{h→0⁻} [f(a+h)−f(a)]/h; similarly for RHD. Differentiability implies continuity, but not vice versa — |x| is continuous at x=0 but not differentiable (LHD = −1, RHD = +1). JEE frequently presents functions and asks at which points they fail to be differentiable: corner points (|x| type), cusp points, and points of discontinuity. The chain rule, product rule, and composite differentiability extend to multi-step questions that combine this chapter with our differentiation guide and our calculus guide.