Limits, Continuity, and Differentiability

Overview

This lecture provides a comprehensive review of limits, continuity, and differentiability, focusing on definitions, key properties, types of discontinuities, and important theorems and strategies for solving problems, especially for competitive exams like JEE.

Limits: Fundamentals and Evaluation Strategies

• The limit of a function at a point investigates the function's behavior near points of indeterminate or undefined value.
• Seven indeterminate forms to remember: 0/0, ∞/∞, 0·∞, ∞–∞, 1^∞, 0^0, ∞^0.
• Limit at x = a exists if both left-hand limit (LHL) and right-hand limit (RHL) exist and are equal and finite.
• Limits can be discussed even if the function is not defined at the point, provided it is defined in a neighborhood.
• Strategies for evaluating limits include substitution, factorization, rationalization, using the binomial theorem, and series expansion.
• Five basic limit laws cover sum, product, constant multiple, division (when denominator ≠ 0), and comparison.
• Special expansions and rationalizations are used for roots and trigonometric expressions in limits.

Important Limits and Golden Rules

• Standard trigonometric and exponential limits:
  • limₓ→0 (sin x)/x = 1, limₓ→0 (tan x)/x = 1, limₓ→0 (1–cos x)/x² = ½, limₓ→0 (eˣ–1)/x = 1.
• For polynomials, if the degree of numerator > denominator, limit is ±∞; if equal, limit is the ratio of leading coefficients.
• Never apply limits partially; always evaluate the whole expression together.
• For greatest integer and fractional part functions, handle limits using specific formulas and value considerations.

Continuity

• A function is continuous at x = a if LHL = RHL = f(a), and a is in the domain.
• Continuity can only be discussed at points in the domain.
• If a function is continuous at a, then the limit exists and equals the function value, but not vice versa.
• For intervals:
  • In (a, b), function must be continuous at every point.
  • In [a, b], must be right-continuous at a and left-continuous at b.

Types of Discontinuity

• Removable: Limit exists but either the function is not defined or f(a) ≠ limit.
  • Isolated point: LHL = RHL ≠ f(a).
  • Missing point: LHL = RHL, f(a) undefined.
• Non-removable: Limit does not exist or LHL ≠ RHL.
  • Jump (finite unequal limits), infinite (limit tends to ∞ or –∞), or oscillatory.

Differentiability

• A function is differentiable at x = a if both left and right derivatives exist, are equal, and finite.
• Differentiability implies continuity, but continuity does not imply differentiability.
• Main causes of non-differentiability: discontinuity, sharp corners (cusp), or vertical tangents.
• Sharp corner: LHD and RHD finite but unequal; vertical tangent: both derivatives tend to ±∞.

Key Terms & Definitions

• Limit — The value a function approaches as the input approaches a given point.
• Left-hand limit (LHL) — Limit as function approaches from left.
• Right-hand limit (RHL) — Limit as function approaches from right.
• Continuity — Function is continuous at a point if limit equals the function value at that point.
• Removable discontinuity — Discontinuity that can be "fixed" by redefining the function.
• Differentiable — A function has a defined tangent (finite slope) at a point.
• Indeterminate Forms — Algebraic expressions where limit cannot be directly deduced.

Action Items / Next Steps

• Review and memorize standard indeterminate forms and limit expansions.
• Practice limit and continuity problems, especially with greatest integer and modulus functions.
• Revisit class notes for series expansions and their uses in limits.
• Complete any unattempted SRQ (Solved Revision Questions) for automatic revision.