Chapter 1: Direct Substitution and Algebraic Methods The first step in any limit problem is direct substitution. If f(a) is a well-defined real number, then lim(x->a) f(x) = f(a) (for algebraic functions). When substitution gives an indeterminate form, algebraic manipulation is the next step.
For polynomial ratios giving 0/0, factor both numerator and denominator. The common factor (x-a) causing 0/0 will cancel. For expressions with surds (square roots), rationalization by multiplying with the conjugate is the standard technique. For differences of nth powers, use x^n - a^n = (x-a)(x^(n-1) + x^(n-2)*a + ... + a^(n-1)).
Chapter 2: Standard Limits and Their Applications Standard limits are pre-computed results that serve as building blocks. The most important are sin(x)/x -> 1, (e^x-1)/x -> 1, and ln(1+x)/x -> 1. These generalize: if f(x) -> 0, then sin(f(x))/f(x) -> 1.
The technique of "adjusting arguments" is crucial. For lim sin(3x)/sin(5x), introduce 3x and 5x in appropriate places: [sin(3x)/(3x)] * [5x/sin(5x)] * (3/5) = 1 * 1 * 3/5 = 3/5.
Chapter 3: The 1^infinity Form Whenever the base approaches 1 and the exponent approaches infinity, use the master formula: lim f(x)^g(x) = e^(lim g(x)*(f(x)-1)). Steps: (1) Verify the 1^infinity form, (2) Compute f(x) - 1, (3) Multiply by g(x), (4) Evaluate the limit, (5) Answer is e raised to that limit.
Chapter 4: L'Hopital's Rule and Taylor Series L'Hopital's Rule transforms the limit of a quotient into the limit of the quotient of derivatives. It requires the original form to be 0/0 or infinity/infinity and the functions to be differentiable.
Taylor series often outperform L'Hopital's Rule. For lim(x->0) (x - sin x)/x^3, Taylor gives the answer in one step (= 1/6), while L'Hopital requires three applications. The strategy is to expand each function to enough terms to resolve the indeterminacy.
Chapter 5: Continuity and Discontinuity Continuity problems in JEE typically involve: (1) Finding parameter values for continuity of piecewise functions, (2) Classifying types of discontinuity, (3) Determining points of discontinuity for composite functions involving [x] or {x}.
The greatest integer function [x] and fractional part {x} are the most commonly tested special functions. Key facts: [x] has jump discontinuity at every integer, {x} has jump discontinuity at every integer, and [f(x)] is discontinuous where f(x) is an integer.