Paper I — Mathematics

Limits, Continuity and Differentiability

Exam-focused notes, visual explanation, traps, revision sheet and every mapped past NDA question.

Your study controls

High yield5 mapped NDA questions

Build the idea

Limits are the gateway to calculus. Start with direct substitution, then factorisation, rationalisation or a standard limit. Continuity compares the left limit, right limit and actual value; differentiability additionally requires matching finite derivatives.

Function, tangent and accumulated area

Concept

A limit describes the value a function approaches near a point. A function is continuous at a point if its limit there equals its value; it is differentiable if its graph has a well-defined tangent (no sharp corner).

Differentiability implies continuity, but not the reverse.

Key Points & Formulas

Standard limits: lim(x→0) sinx/x = 1; lim(x→0)(eˣ−1)/x = 1.
For 0/0 forms, factorise or use L'Hôpital's idea.
Continuous at x=a: LHL = RHL = f(a).
|x| is continuous everywhere but not differentiable at x = 0.

Worked Example

Q. Evaluate lim(x→3) (x² − 9)/(x − 3).

Direct substitution gives 0/0, so factorise.

(x²−9)/(x−3) = (x−3)(x+3)/(x−3) = x+3.

Limit = 3 + 3 = 6.

Common traps

Cancelling a factor before stating the point is excluded.
Using lim sin x/x = 1 when the angle is in degrees.
Assuming continuity implies differentiability.

NDA shortcuts

For sin(kx)/x, multiply and divide by k.
For radicals, multiply by the conjugate.
At piecewise joins, compare LHL, RHL and f(a) in that order.

One-screen revision sheet

Recall before you practise

limₓ→0 sin x/x = 1
limₓ→0 (eˣ−1)/x = 1
Differentiable ⇒ continuous

Past-paper bank

Practise every mapped question

Filter by year

5 NDA questions

Choose an answer before the solution appears. Option order is mixed automatically.

Question 1

NDA I 2026

What is the derivative of $\dfrac{x}{|x|}$ with respect to $x$ , where $x \lt 0$ ?

Question 2

NDA II 2025

What is $\lim\limits_{x \to 0^-} f(x)$ equal to?

Question 3

NDA II 2024

Which one of the following is correct regarding $\footnotesize \displaystyle\lim_{x \to 3} \dfrac{|x-3|}{x-3}$ ?

Question 4

NDA II 2024

What is $\footnotesize \displaystyle\lim_{\theta \to \pi/2}(\sec\theta - \tan\theta)$ equal to?

Question 5

NDA I 2024

What is lim(x→0−) h(x) + lim(x→0+) h(x) equal to?

← Three-Dimensional Geometry Continue in Test Module Differentiation →