Limits

Definition of Limit

• The real number l is the limit of a function f(x) as x → a if for every ε > 0, there exists δ > 0 such that:
• 0 < |x − a| < δ ⇒ |f(x) − l| < ε
• Denoted by: lim_x→a f(x) = l

Indeterminate Forms

• 0/0
• ∞/∞
• 0 × ∞
• ∞ − ∞
• ∞⁰, 0⁰, 1⁰
• Limits are evaluated to resolve these forms

Left Hand Limit (LHL) & Right Hand Limit (RHL)

• LHL: lim_x→a⁻ f(x) = lim_h→0⁺ f(a − h)
• RHL: lim_x→a⁺ f(x) = lim_h→0⁺ f(a + h)
• Limit exists at x = a if LHL = RHL
• If limit exists, it is unique

Properties of Limits

• If lim_x→a f(x) = l and lim_x→a g(x) = m, then:
• Sum: lim(f ± g) = l ± m
• Product: lim(fg) = lm
• Quotient: lim(f/g) = l/m (m ≠ 0)
• Constant multiple: lim(Kf) = K · lim f
• Modulus: lim|f(x)| = |lim f(x)|
• Power rule: lim[f(x)]ⁿ = [lim f(x)]ⁿ
• Composite function: lim f(g(x)) = f(lim g(x))

Important Standard Limits

• lim_x→0 (sin x)/x = 1
• lim_x→0 (tan x)/x = 1
• lim_x→0 (1 − cos x)/x² = 1/2
• lim_x→0 (eˣ − 1)/x = 1
• lim_x→0 (aˣ − 1)/x = ln a, a > 0
• lim_x→0 ln(1 + x)/x = 1
• lim_x→0 (1 + x)^1/x = e
• lim_x→∞ (1 + 1/x)^x = e

Algebraic Limits

• lim_x→a (xⁿ − aⁿ)/(x − a) = n·aⁿ⁻¹
• lim_x→a (xᵐ − aᵐ)/(xⁿ − aⁿ) = (m/n)·aᵐ⁻ⁿ

Methods of Evaluating Limits

• Substitution method
• Factorisation method
• Rationalisation / Double rationalisation
• Expansion method (using series)
• When x → ∞ (highest power method)
• Simplification
• L’Hospital’s Rule
• Sandwich (Squeeze) Theorem

L’Hospital’s Rule

• Applicable only for forms 0/0 or ∞/∞
• If lim f(x) = 0 and lim g(x) = 0 or both → ∞:
• lim f(x)/g(x) = lim f′(x)/g′(x) (if RHS exists)
• Differentiate numerator and denominator separately

JEE Main Focus Tips

• Standard limits must be memorised
• Check LHL and RHL for piecewise functions
• Convert complex expressions to standard forms
• Use expansion for small x problems
• L’Hospital’s Rule only after confirming 0/0 or ∞/∞ form