How to Calculate Limits: Methods, Rules, and Examples

Limits are the foundation upon which calculus is built. They describe the value a function approaches as its input gets closer and closer to a particular point. Understanding limits is essential before tackling derivatives or integrals.

This guide covers the core techniques for evaluating limits, including direct substitution, factoring, rationalisation, and L'Hopital's rule, with worked examples for each.

What Is a Limit?

The limit of as approaches is the value that gets arbitrarily close to. We write:

The limit exists only if the function approaches the same value from both sides. If the left-hand and right-hand limits differ, the two-sided limit does not exist.

Method 1: Direct Substitution

The simplest approach is to substitute directly into the function. If this produces a real number (not an indeterminate form), you have your answer.

Example 1: Direct Substitution

Evaluate .

Since substitution gives a finite value, the limit is .

Method 2: Factoring

When direct substitution produces , the expression is indeterminate. Often you can factor the numerator and denominator to cancel the common factor causing the problem.

Example 2: Factoring

Evaluate .

Direct substitution gives . Factor the numerator:

Now substitute:

Method 3: Rationalisation

When the expression involves square roots, multiplying by the conjugate can eliminate the indeterminate form.

Example 3: Rationalisation

Evaluate .

Multiply numerator and denominator by the conjugate :

Cancel and substitute:

Method 4: L'Hopital's Rule

When substitution gives or , L'Hopital's rule allows you to differentiate the numerator and denominator separately and re-evaluate:

This is valid provided the new limit exists (or is infinite).

Example 4: L'Hopital's Rule

Evaluate .

Direct substitution gives . Apply the rule:

Example 5: Repeated Application

Evaluate .

This gives . Apply L'Hopital's rule once:

Still , so apply it again:

Limits at Infinity

To find the behaviour of a rational function as , divide every term by the highest power of in the denominator. Terms with in the denominator vanish.

Example 6: Limit at Infinity

Evaluate .

Divide numerator and denominator by :

Key Points to Remember

1. Always try direct substitution first. It is the quickest method and works whenever the function is continuous at the point.
2. Indeterminate forms require further work. The forms and do not mean the limit is zero or undefined.
3. L'Hopital's rule is powerful but not universal. It only applies to indeterminate quotients, not to all limit problems.

Try It Yourself

Practise evaluating limits with our free Limit Calculator. Enter any function and see the step-by-step evaluation, including the technique used.