Limits, Continuity & Differentiability

Concept overview

A limit lim_{x → a} f(x) = L means f(x) can be made arbitrarily close to L by taking x sufficiently close to a. The ε-δ definition makes this precise, but for JEE you rarely need the formal proof. What matters is fluency with standard limits: sin(x)/x → 1, (1 − cos x)/x² → ½, (aˣ − 1)/x → ln a, (1 + x)^(1/x) → e.

L'Hôpital's rule: for indeterminate forms 0/0 or ∞/∞, lim f/g = lim f'/g'. Repeat if the new form is still indeterminate. Taylor-series expansion is another way to resolve indeterminates — expand numerator and denominator to enough terms and cancel.

Continuity at x = a requires three things: f(a) is defined, lim_{x → a} f(x) exists, and the two are equal. Discontinuities come in three types: removable (limit exists but ≠ f(a)), jump (left and right limits differ), and infinite (limit is ±∞).

Differentiability at a point implies continuity there but not vice versa. |x| is continuous at 0 but not differentiable (left and right derivatives differ). Piece-wise defined functions require checking continuity and differentiability at the joining points.

Key formulas & concepts

• Standard limit: lim_{x→0} sin x / x = 1
• L'Hôpital's rule: lim f/g = lim f'/g' for 0/0 or ∞/∞
• Euler's limit: lim_{x→0} (1 + x)^(1/x) = e
• Continuity condition: f(a) exists, lim_{x→a} f(x) exists, lim = f(a)
• Left/right derivative: differentiable ⇔ LHD = RHD at that point
• Differentiable ⇒ continuous: converse is false

Common mistakes to avoid

• Applying L'Hôpital to non-indeterminate forms — you get a wrong answer.
• Treating |x| as differentiable at 0 — it isn't; LHD = −1 ≠ RHD = 1.
• Forgetting to check both sides for piece-wise continuity.
• Assuming continuity implies differentiability — many continuous functions are not differentiable.

JEE Main relevance

JEE Main asks 2 questions every paper — usually one limit evaluation and one continuity/differentiability of a piece-wise function.

JEE Advanced relevance

Advanced problems test differentiability at multiple joining points or use asymptotic expansions for tricky limits. Functional equations combined with continuity are also common.

Previous-year questions (PYQ)

Piece-wise continuity/differentiability problems dominate the PYQ pool for this chapter.