Mathematics Limit, Continuity and Differentiability JEE Syllabus

Limit, Continuity and Differentiability is a part of calculus where you start understanding how Functions behave when their input values change. Instead of only calculating outputs, you begin to observe how a Function behaves near a point, whether it moves smoothly, and how its rate of change behaves. This chapter acts as a bridge between basic algebra and advanced calculus because it introduces the idea of studying behaviour rather than just values.

You gradually move from understanding simple Limits to checking whether a Function is continuous and then exploring how smoothly it can change. These ideas help you analyse Functions in a more meaningful way and are widely used in physics and other areas where motion and change are studied.

Concept of Limit

A Limit describes the value a Function approaches as the input gets closer to a particular point. It focuses on behaviour near a point rather than the actual value at that point.

If a Function approaches the same value from both sides (left and right), the Limit exists. Limits are the foundation for defining Continuity and Differentiability.

Continuity of a Function

A Function is continuous at a point if there is no break, jump or hole in its graph at that point.

Mathematically, a Function f(x)f(x)f(x) is continuous at x=cx = cx=c if:

• Left-hand Limit exists

• Right-hand Limit exists

• Both are equal to each other and equal to f(c)f(c)f(c)

So,

lim⁡x→c−f(x)=lim⁡x→c+f(x)=f(c)\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = f(c)x→c−lim​f(x)=x→c+lim​f(x)=f(c)

Continuity ensures that the Function behaves smoothly without sudden changes.

On a closed interval [a,b][a, b][a,b], Continuity must hold at every point, including endpoints, using one-sided Limits.

Types of Discontinuity

When a Function is not continuous, it is called discontinuous. Discontinuity occurs when Limits do not match or when the Function is not defined properly.

Common types:

• Removable Discontinuity: A hole in the graph where the Limit exists, but the Function value is not defined or mismatched.

• Jump DisContinuity: Left-hand and right-hand Limits exist but are not equal.

• Infinite DisContinuity: Function grows without bound near a point.

Algebra of Continuous Functions

Continuous Functions behave nicely under operations.

If f(x)f(x)f(x) and g(x)g(x)g(x) are continuous at a point ccc, then:

• f+gf + gf+g is continuous

• f−gf - gf−g is continuous

• f⋅gf \cdot gf⋅g is continuous

• fg\frac{f}{g}gf​ is continuous (if g(c)≠0g(c) \neq 0g(c)=0)

This helps in constructing more complex continuous Functions from simpler ones.

Standard Continuous Functions

Some Functions are always continuous in their domain:

• Polynomial Functions: Continuous everywhere

• Trigonometric Functions: sin⁡x\sin xsinx and cos⁡x\cos xcosx are continuous for all real numbers

• Rational Functions: Continuous except where the denominator is zero

• Exponential Functions: Continuous for all real values

These properties are widely used in calculus problems.

Differentiability

A Function is differentiable at a point if its derivative exists there.

Differentiability means the Function has a well-defined tangent (slope) at that point without sharp corners or breaks.

Every differentiable Function is continuous, but every continuous Function is not necessarily differentiable. For example, ∣x∣|x|∣x∣ is continuous everywhere but not differentiable at x=0x = 0x=0.

Implicit Differentiation

Sometimes a Function cannot be written clearly as y=f(x)y = f(x)y=f(x). In such cases, we use implicit differentiation.

Here, both variables are differentiated with respect to xxx, treating yyy as a Function of xxx.

This method is useful for equations like:

• x2+xy+y2=0x^2 + xy + y^2 = 0x2+xy+y2=0

• Complex trigonometric or mixed-variable expressions

It helps find dydx\frac{dy}{dx}dxdy​ even when yyy is not isolated.

Exponential and Logarithmic Differentiation

In this topic, you will learn how exponential and logarithmic Functions behave when they are differentiated and how these rules help you handle expressions that look complicated at first. You will understand how exponential Functions follow a simple and consistent pattern under differentiation, and how logarithmic Functions convert multiplicative expressions into simpler additive or subtractive forms that are easier to work with. 

Logarithmic Differentiation

Used when both base and exponent are variable, like y=[u(x)]v(x)y = [u(x)]^{v(x)}y=[u(x)]v(x).

Steps:

• Take the natural log on both sides

• Use log rules to simplify powers

• Differentiate using the chain and product rules

• Multiply back by the original Function

This simplifies otherwise complex differentiation problems.

Higher Order Derivatives

The second derivative is obtained by differentiating the first derivative again.

d2ydx2\frac{d^2y}{dx^2}dx2d2y​

It gives information about curvature:

• Positive second derivative → concave upward

• Negative second derivative → concave downward

For trigonometric Functions, repeated differentiation often produces repeating patterns, showing cyclic behaviour.