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.

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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

Worked Example

Evaluate the following limit by applying the limit properties:

\lim_{x \to 2} \dfrac{x^2 + 3x}{5x - 1}

Step 1: Apply the Quotient Rule - We treat the numerator and denominator as separate functions:

\dfrac{\lim_{x \to 2} (x^2 + 3x)}{\lim_{x \to 2} (5x - 1)}

Step 2: Apply the Sum and Difference Rules - Break down the terms within the numerator and denominator:

\dfrac{\lim_{x \to 2} x^2 + \lim_{x \to 2} 3x}{\lim_{x \to 2} 5x - \lim_{x \to 2} 1}

Step 3: Apply Constant Multiple Rule and Power Rule:

\dfrac{(\lim_{x \to 2} x)^2 + 3(\lim_{x \to 2} x)}{5(\lim_{x \to 2} x) - 1}

Step 4: Substitute the value x = 2:

\dfrac{(2)^2 + 3(2)}{5(2) - 1} = \dfrac{4 + 6}{10 - 1} = \dfrac{10}{9}

Practice Questions & Solutions

Score:

Given that

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and

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find the value of

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Solution

Applying the Sum and Constant Multiple Rules:
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Substitute the given values:
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Evaluate the following limit:

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Solution

Apply the Quotient Rule and evaluate:

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Find the limit of the constant function:

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Solution

The limit of a constant is always the constant itself:

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Evaluate the limit using the Product Rule:

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Solution

Distribute the limit across the product:

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Solve the following limit where direct substitution is possible:

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Solution

Apply the Quotient Rule:

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Evaluate the following limit:

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Solution

Direct substitution yields:

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which is an indeterminate form.

Factorise the numerator:

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Simplify the expression for $\text{[math]}$ :

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Apply Direct Substitution:

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Evaluate the following limit:

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Solution

Direct substitution yields

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Multiply the numerator and denominator by the conjugate $\text{[math]}$

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Expand the numerator using:

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Cancel x from the numerator and denominator:

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Substitute x = 0:

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Given that

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and

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Evaluate:

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Solution

Apply the Quotient, Sum, Power, and Root rules individually.

Evaluate the numerator limit:

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Evaluate the denominator limit:

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Combine results:

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Evaluate the following trigonometric limit:

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Solution

Recall the standard limit:

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Adjust the expression to isolate

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Apply the Constant Multiple Rule:

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Substitute the standard limit value:

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Evaluate the following limit as x approaches infinity:

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Solution

Direct evaluation leads to

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Divide every term by the highest power of $x$ in the denominator $\text{[math]}$

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Apply the rule

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for any n > 0:

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Gianpiero Placidi

UK-based Chemistry graduate with a passion for education, providing clear explanations and thoughtful guidance to inspire student success.

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Chapters

Theory
Worked Example
Practice Questions & Solutions

Understanding the Properties of Limits

Theory

The Fundamental Limit Laws

Direct Substitution and Continuity

Worked Example

Practice Questions & Solutions

Theory

Exponent Rules

Composition of Functions

Constant Functions

Convergent and Divergent Sequences

Even and Odd Functions

Exponential Function

Graphing Parabolas

Arithmetic Sequence

Mean Value Theorem

Rolle’s Theorem

Intervals of Increase and Decrease

Periodic Functions

Graphing rational function I

Graphing rational function III

Inflection Points

Increasing and Decreasing Functions

Piecewise Functions

Trigonometric Functions

Linear Function

Domain of a Function

Maxima, Minima and Inflection Points

Linear Functions Worksheet

Monotone Sequence

Radical Functions

Function

Geometric Sequence

Coordinates of X Intercept Explained: GCSE and A Level

Bounded Sequence: Monotonic and Non-Monotic

L’Hôpital’s Rule

Graphing polynomial function I

Types of Functions

Coordinates in the Plane

Optimization

nth Term

Graphing rational function V

Limit of a Sequence

Absolute Value Function

Sequences

Geometric Sequence Problems And Solutions

Arithmetic Sequence Problems with Solutions

Logarithmic Functions

Concave and Convex Functions: Theory and Examples

Rational Functions: Graphs, Hyperbolas and Asymptotes Explained

Bounded Function Examples: Theory and Practice Questions

Maxima and Minima Examples With Solutions

Asymptote Functions: Theory and Practice Problems

Absolute Maxima Explained: Master Max/Min On Intervals

Inverse Function Explained: Simple Steps for A-Level Maths

Graphs of Quadratic Equations: Practice Questions and Answers

Applying Limits in Math: Limit Laws Made Clear

Formulas

Sequence Formulas | Arithmetic, Geometric and More

Exercises

Inflection Points Worksheet

Sequences Problems

Concavity and Convexity Worksheet

Increase and Decrease Worksheet

Limit of Sequence Problems

Applications of Derivatives Worksheet

Inverse Functions Worksheet

Exponential, Logarithmic and Trigonometric Functions Worksheet

Composition of Functions Worksheet

Graphing Worksheet

Mean Value Theorem Word Problems

Maximum and Minimum Word Problems

Quadratic Function Word Problems

Rational Functions Worksheet

Domain Worksheet

L’Hopital’s Rule Problems

Even and Odd Functions Worksheet

Asymptotes Worksheet

Absolute Value Functions Worksheet

Piecewise Functions Worksheet