Calculating Limits Using the Limit Laws

Limit laws are a set of fundamental rules in calculus that allow you to find the limit of complex functions by breaking them down into simpler parts. These laws are based on the properties of limits and provide a systematic way to evaluate limits algebraically.

Limit laws are used because they provide a rigorous and efficient way to evaluate limits algebraically, without relying solely on graphs or tables of values. They allow us to determine the limit of sums, differences, products, quotients, and powers of functions if the limits of the individual functions exist.

Assuming limx→c f(x) = L and limx→c g(x) = M exist, and 'k' is a constant:

• **Sum Law:** limx→c [f(x) + g(x)] = L + M
• **Difference Law:** limx→c [f(x) - g(x)] = L - M
• **Constant Multiple Law:** limx→c [k * f(x)] = k * L
• **Product Law:** limx→c [f(x) * g(x)] = L * M
• **Quotient Law:** limx→c [f(x) ÷ g(x)] = L ÷ M (provided M ≠ 0)
• **Power Law:** limx→c [f(x)]n = Ln (where n is a positive integer)
• **Root Law:** limx→c √[f(x)] = √L (where n is a positive integer, and if n is even, L must be > 0)

While limit laws are powerful, they require that the limits of the individual parts exist. You might need other techniques if direct substitution leads to indeterminate forms like 0/0 or ∞/∞. These techniques include:

• **Factoring and Cancellation:** For rational functions.
• **Multiplying by the Conjugate:** Often used with square roots.
• **L'Hôpital's Rule:** When you have indeterminate forms 0/0 or ∞/∞ (requires derivatives).
• **Using Special Limits:** Like limx→0 (sin x ÷ x) = 1.
• **Considering One-Sided Limits:** Especially for piecewise functions or functions with vertical asymptotes.

Often, algebraic manipulation is used to transform the function into a form where limit laws *can* be applied.