Limits, Continuity and Differentiability
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Direct answer
A function f is continuous at a point a when the left limit, right limit and the value f(a) all agree, and differentiable at a when the difference quotient [f(a + h) - f(a)]/h approaches the same finite value from both sides. Differentiability implies continuity, but the converse fails, with |x| at 0 the standing counterexample. Limits are evaluated by matching against a short list of standard forms, with L'Hopital's rule reserved for genuine 0/0 or infinity/infinity forms.
What you must remember
- Standard limits as x → 0: sin x / x → 1, tan x / x → 1, (1 - cos x)/x^2 → 1/2; (1 + x)^(1/x) → e; (e^x - 1)/x → 1; (a^x - 1)/x → ln a; ln(1 + x)/x → 1; [(1 + x)^n - 1]/x → n. All trigonometric limits assume radians.
- One-to-infinity form: if f → 1 and g → infinity, then f^g → e^(limit of (f - 1) g) — the most used trick in the chapter.
- Continuity at a: left limit = right limit = f(a); for piecewise functions, match branches and the assigned value to solve for unknown constants.
- Differentiability: left and right derivatives must agree; differentiable implies continuous; continuous does not imply differentiable — |x| at 0 is the standing counterexample.
- Algebra of continuity: sums, products, quotients (non-zero denominator) and composites of continuous functions are continuous; discontinuity enters at joints, denominators and log or sqrt boundaries.
- Rolle's theorem: if f is continuous on [a, b], differentiable on (a, b) and f(a) = f(b), then f'(c) = 0 for some c in (a, b); LMVT replaces the zero by the average slope (f(b) - f(a))/(b - a).
- L'Hopital's rule applies only to 0/0 or infinity/infinity forms and only when the new limit exists; check the form before differentiating.
Common confusion
Two habits wreck otherwise correct work. First, quoting sin x/x → 1 in degree mode or after substituting x → pi instead of 0 — the standard limit is tied to radians and to zero. Second, treating 1^infinity as 1: the base tends to 1 but the exponent grows, and the answer is e raised to a computable limit. With modulus and floor functions, compute both one-sided limits before declaring anything — a formula that exists is not automatically continuous.
Exam-focused takeaway
JEE Main asks limit evaluation, parameter matching for continuity, and differentiability of |f(x)| or max-min expressions at their joints — mostly numerical-value questions. JEE Advanced probes non-existence arguments, functions built from integrals or recursion, and Rolle/LMVT applied to constructed functions to prove roots or inequalities. Classify the form first, choose the standard tool second, and verify one-sided statements whenever modulus or floor functions appear.
Frequently asked questions
Is every continuous function differentiable?
No. |x| is continuous everywhere but not differentiable at 0, since its left and right derivatives are -1 and 1.
How do I evaluate a one-to-infinity limit?
Write it as e^(limit of (f - 1) × g) where f is the base tending to 1 and g the exponent tending to infinity.
What is the limit of (1 - cos x)/x^2 as x → 0?
1/2 — write 1 - cos x = 2 sin^2(x/2) and use sin t/t → 1.
When may L'Hopital's rule be used?
Only for 0/0 or infinity/infinity forms, and only when the limit of the quotient of derivatives exists; check the form first.
What does Rolle's theorem conclude?
Under continuity on [a, b], differentiability on (a, b) and equal end values, some interior c has f'(c) = 0 — the standard tool for proving an equation has a root.