Limits and continuity form the conceptual bedrock upon which all of differential and integral calculus is built. In JEE Main, this topic carries a weightage of 3-4 questions per year, making it one of the most important foundational topics in mathematics.

The concept of a limit captures the idea of approaching a value. Formally, lim(x->a) f(x) = L means that for every epsilon > 0, there exists a delta > 0 such that whenever 0 < |x - a| < delta, we have |f(x) - L| < epsilon. For practical purposes in JEE, we work with left-hand limits (LHL) and right-hand limits (RHL). The two-sided limit exists if and only if LHL equals RHL.

When direct substitution leads to an indeterminate form, we need specialized techniques. The seven indeterminate forms are: 0/0, infinity/infinity, 0 times infinity, infinity minus infinity, 0^0, 1^infinity, and $infinity^0$. Each form has preferred resolution methods.

For 0/0 forms, the hierarchy of techniques is: (1) factorization for polynomial expressions, (2) rationalization for expressions with square roots, (3) standard limits for trigonometric and exponential expressions, (4) Taylor/Maclaurin expansions for complex combinations, and (5) L'Hopital's Rule as the universal fallback.

The standard limits that must be memorized include: sin$\frac{x}{x}$ -> 1, tan$\frac{x}{x}$ -> 1, $\frac{e^x - 1}{x}$ -> 1, ln$\frac{1+x}{x}$ -> 1, $\frac{a^x - 1}{x}$ -> ln(a), (1+x)^$\frac{1}{x}$ -> e, $\frac{1-cos x}{x}$^2 -> 1/2, and $\frac{x^n - a^n}{(x - a)}$ -> n*a^(n-1).

The 1^infinity form deserves special attention as it appears in virtually every JEE paper. The resolution formula is: lim f(x)^g(x) = e^(lim g(x)[f(x)-1]) when f(x) -> 1 and g(x) -> infinity.

L'Hopital's Rule states that for 0/0 or infinity/infinity forms, lim f$\frac{x}{g}$(x) = lim f'$\frac{x}{g}$'(x), provided the conditions of differentiability are met and the right-side limit exists. A critical error to avoid is using the quotient rule instead of differentiating numerator and denominator separately.

Taylor/Maclaurin expansions provide an efficient alternative: sin(x) = x - $x^3$/6 + $x^5$/120 - ..., cos(x) = 1 - $x^2$/2 + $x^4$/24 - ..., $e^x$ = 1 + x + $x^2$/2 + $x^3$/6 + ..., ln(1+x) = x - $x^2$/2 + $x^3$/3 - ... The strategy is to expand only enough terms to cancel the indeterminacy.

Continuity at a point requires three simultaneous conditions: the function must be defined at the point, the limit must exist, and the limit must equal the function value. Discontinuities are classified as removable (limit exists but value is wrong), jump (LHL != RHL), infinite (limit is infinite), or oscillatory (limit doesn't exist due to oscillation).

The Intermediate Value Theorem states that a continuous function on a closed interval takes every value between its endpoint values. This is the primary tool for proving existence of roots. The Sandwich Theorem is essential for handling oscillatory limits.

Key traps include: sqrt($x^2$) = |x| (not x when x is negative), sin$\frac{x}{x}$ = 1 only in radians, [sin$\frac{x}{x}$] = 0 near x = 0 (not 1), and the fractional part function {x} approaching 1 from the left of any integer.