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Introduction

Calculus is essentially the study of change, and at the heart of calculus lies the concept of limits. Whether you are finding the gradient of a curve or the area under a graph, you are using limits. Understanding the Properties of Limits, often called Limit Laws, allows us to break down complex algebraic expressions into manageable parts, making it significantly easier to evaluate the behaviour of functions as they approach specific values.

Theory

To work effectively with limits, we assume that the limits of two individual functions, f(x) and g(x), both exist as x approaches a value a. Let c be a constant. The following rules are the fundamental tools used in A-Level calculus.

The Fundamental Limit Laws

Sum Rule - The limit of a sum is the sum of the limits:

\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)

Difference Rule - The limit of a difference is the difference of the limits:

\lim_{x \to a} [f(x) - g(x)] = \lim_{x \to a} f(x) - \lim_{x \to a} g(x)

Constant Multiple Rule - A constant can be moved outside the limit:

\lim_{x \to a} [c \cdot f(x)] = c \cdot \lim_{x \to a} f(x)

Product Rule - The limit of a product is the product of the limits:

\lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)

Quotient Rule - The limit of a quotient is the quotient of the limits, provided the denominator is not zero:

\lim_{x \to a} \dfrac{f(x)}{g(x)} = \dfrac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)} \text{ if } \lim_{x \to a} g(x) \neq 0

Power Rule - The limit of a function raised to a power is the limit of that function, all raised to that power:

\lim_{x \to a} [f(x)]^n = \left[ \lim_{x \to a} f(x) \right]^n

Root Rule - The limit of a root of a function is the root of the limit of the function:

\lim_{x \to a} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x \to a} f(x)}

Direct Substitution and Continuity

If a function is continuous at a point a, we can find the limit simply by substituting the value into the function. This is known as Direct Substitution.

\lim_{x \to a} f(x) = f(a)

However, if direct substitution results in an indeterminate form such as 0÷0, we must use algebraic manipulation (like factoring) or specific limit properties to find the solution.

Method	When to Use	Action
Direct Substitution	Continuous functions	Plug the value in directly
Factoring	Polynomial fractions	Cancel out common terms causing 0/0
Rationalisation	Functions with roots	Multiply by the conjugate