Operations on limits and indeterminate forms

Operational Rules

If lim(x→a) f(x) = L and lim(x→a) g(x) = M (where L and M are finite), then:

• lim (f+g) = L+M
• lim (fg) = LM
• lim (f/g) = L/M, if M ≠ 0

These rules fail as soon as L or M is infinite or zero in certain combinations: these are the indeterminate forms.

The four indeterminate forms

Form	Typical example
0/0	lim(x→0) sin(x)/x
∞/∞	lim(x→+∞) (x²+1)/(3x²)
∞ – ∞	lim(x→∞+) (√(x+1) – √(x))
0 × ∞	lim(x→0+) x × ln(x)

Methods for resolving indeterminacy

• Factorisation and simplification (e.g. (x² - 1)/(x - 1) = x + 1 for x ≠ 1)
• Conjugate quantities for roots
• Factoring out the dominant term in ±∞
• Change of variables
• Use of equivalents or limit expansions

Example: lim(x → +∞) (√(x + 1) − √(x)) = lim (x + 1 − x)/(√(x + 1) + √(x)) = lim 1/(√(x + 1) + √(x)) = 0.

Common pitfall

L’Hôpital’s rule (derivative of f divided by derivative of g) applies ONLY to the forms 0/0 or ∞/∞, and only if the derivatives are well-defined in the neighbourhood of the point; applying it blindly to a non-indeterminate form yields an incorrect result.