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$\frac{x}{x}$ -> 1, $\frac{e^x-1}{x}$ -> 1, and ln$\frac{1+x}{x}$ -> 1. These generalize: if f(x) -> 0, then sin(f(x))/f(x) -> 1.

The technique of "adjusting arguments" is crucial. For lim sin$\frac{3x}{sin}$(5x), introduce 3x and 5x in appropriate places: [sin$\frac{3x}{(3x)}$] * [5x/sin(5x)] * $\frac{3}{5}$ = 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) $\frac{x - sin x}{x}$^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.