Limits and Continuity for JEE: L'Hôpital's Rule and Standard Limits Guide

L'Hôpital's rule is valid only for 0/0 and ∞/∞ indeterminate forms. First verify the form, then differentiate numerator and denominator separately. Can be applied repeatedly, but check the form each time.

Removable discontinuity: the limit exists but ≠ f(a) or f(a) is undefined. It can be removed by redefining f(a). Non-removable: the limit doesn't exist (jump or infinite discontinuity).

For lim f^g where f→1 and g→∞: use lim f(x)^g(x) = e^(lim g(x)·(f(x)-1)). This converts 1^∞ to e^(∞×0) form which can be solved with L'Hôpital after conversion.

The limit of f(x) as x → a describes what value f(x) approaches as x gets closer to a (from either side). The value f(a) is what the function actually equals at x = a. They are equal if the function is continuous at a.

Left-hand limit: lim(x→a⁻) f(x) — as x approaches a from values less than a. Right-hand limit: lim(x→a⁺) f(x) — as x approaches a from values greater than a. The two-sided limit exists only if both one-sided limits exist and are equal.

A limit fails to exist when: (1) left and right limits differ, (2) the function approaches infinity (±∞), (3) the function oscillates rapidly (like sin(1/x) as x→0), (4) the function doesn't approach any particular value.

The squeeze theorem: if g(x) ≤ f(x) ≤ h(x) for all x near a (except possibly at a), and lim g(x) = lim h(x) = L, then lim f(x) = L. Used for limits like lim(x→0) x² sin(1/x).

For rational functions: compare degrees of numerator and denominator. If degree num < degree den: limit = 0. If equal: limit = coefficient ratio. If greater: limit = ±∞ (or use L'Hôpital). Divide numerator and denominator by highest power of x.

A function is continuous at a if lim f(x) = f(a). A function is differentiable at a if f'(a) exists (the derivative). Differentiability implies continuity, but continuity does NOT imply differentiability (e.g., |x| at x=0 is continuous but not differentiable).

f(x) is continuous at x = a if: (1) f(a) is defined, (2) lim(x→a) f(x) exists (both one-sided limits equal), (3) lim(x→a) f(x) = f(a). If any condition fails, the function is discontinuous at a.

Removable: limit exists but doesn't equal f(a) (hole). Jump: left and right limits differ (step). Infinite: limit is ±∞ (vertical asymptote). Oscillating: function oscillates (no limit).

lim(x→0) sin x/x = 1. This is proven by the squeeze theorem: cos x < sin x/x < 1 for small x. This is one of the most important standard limits in JEE. It is also the basis for the derivative of sin x.

If f is differentiable at a, then f is continuous at a. But the converse is false: f(x) = |x| is continuous at 0 but not differentiable at 0. A function can be continuous everywhere but differentiable nowhere (Weierstrass function).