How do I find a limit when I get 0/0?

Cancel the factor that causes the zero. Factorise polynomials, multiply by the conjugate for square roots, or rewrite the expression so that sin θ/θ or (eˣ − 1)/x appears, then substitute.

Is the epsilon-delta definition of a limit required in DSE M2?

No. The curriculum asks only for the intuitive concept of a limit; the rigorous definition and proofs of the limit theorems are not required.

Can a limit fail to exist?

Yes. For example 1/x has no limit as x tends to 0, because the values grow without bound and have opposite signs on the two sides.

Limits (DSE M2): How to Find the Limit of a Function

The limit of f(x) as x → a is the value f(x) approaches, whether or not f(a) exists. If substituting gives 0/0, simplify first: factorise, rationalise, or use the standard limits lim sin θ/θ = 1 (θ in radians) and lim (eˣ − 1)/x = 1 as the variable tends to 0.

Solve Limits questions with AI

y = sin x / x near x = 0. The function is undefined at 0, yet the curve heads to the height 1 from both sides. Try it yourself →

Limit Formulas (DSE M2)

Trigonometric limitlim_θ→0 (sin θ)/θ = 1 (θ in radians)

Exponential limitlim_x→0 (eˣ − 1)/x = 1

Sum, product, quotientThe limit of a sum, product or quotient is the sum, product or quotient of the limits, provided each limit exists (and the denominator’s limit is not 0)

Rational function at infinityDivide the numerator and denominator by the highest power of x

SurdsMultiply by the conjugate, e.g. √(x + 2) − √x = 2 / (√(x + 2) + √x)

Worked Examples — Limits

Example 1: A 0/0 limit by rationalisation

Find lim_x→0 (√(x + 4) − 2) / x.

Substituting gives 0/0, so multiply top and bottom by √(x + 4) + 2
(√(x + 4) − 2)(√(x + 4) + 2) = (x + 4) − 4 = x
(√(x + 4) − 2) / x = 1 / (√(x + 4) + 2)
As x → 0: 1 / (2 + 2) = 1/4

Answer: 1/4

Example 2: Using lim sin θ / θ = 1

Find lim_x→0 (1 − cos x) / x².

1 − cos x = 2 sin²(x/2)
(1 − cos x) / x² = 2 sin²(x/2) / x² = ½ · [sin(x/2) / (x/2)]²
As x → 0, sin(x/2) / (x/2) → 1
½ × 1² = ½

Answer: 1/2

💡 Exam Traps & Secrets

⚠️ 0/0 is not an answerGetting 0/0 means "simplify first", not "the limit is 0" or "the limit does not exist". Factorise, rationalise or use a standard limit, then substitute again.

⚠️ sin θ / θ → 1 needs radiansThe standard limit holds only when θ is in radians, and the same expression must appear inside the sine and in the denominator: sin 3x / 3x → 1, so sin 3x / x → 3.

✏️ Self-Test: Quick Interactive Practice

How to study: Click any question below to instantly load it on the coordinate grid and check your steps.

(Factorising)Find the limit of (x² − 9)/(x − 3) as x tends to 3.

(Standard trig limit)Find the limit of sin 5x / (2x) as x tends to 0.

(Limit at infinity)Find the limit of (3x² − x + 1)/(2x² + 5) as x tends to infinity.