Class 11 Maths Notes

Limits and Derivatives Class 11 Notes

Complete, exam-ready notes on limits and derivatives: the intuitive idea of a limit, left-hand and right-hand limits, algebra of limits, standard limits, continuity, the derivative as a limit from first principle, algebra of derivatives, and derivatives of standard functions — written for CBSE boards and JEE/NEET revision.

Class11SubjectMathematicsCoversCBSE · JEE

Written byDeep Narayan· Science & Mathematics EducatorReviewed byPushpanjali

What is the main idea of limits and derivatives?

A limit describes the value a function approaches; the derivative measures the instantaneous rate of change — the slope of the tangent — defined as a limit of the difference quotient.

Intuitive Idea of a Limit

Limit

As x approaches a value a from either side, if f(x) gets arbitrarily close to a unique number L, we say the limit of f(x) as x tends to a is L, written $\lim_{x \to a} f(x) = L$ . The function need not be defined at x = a for the limit to exist.

The limit exists only when the left-hand and right-hand limits are equal.
Graphically, the limit is the y-value the curve approaches, regardless of whether the point is plotted.
Limits form the foundation for both continuity and differentiation.

Left-Hand and Right-Hand Limits

Left-hand limit (LHL) and right-hand limit (RHL)

The left-hand limit $\lim_{x \to a^-} f(x)$ is the value f(x) approaches as x approaches a from below. The right-hand limit $\lim_{x \to a^+} f(x)$ is from above. The limit exists if and only if LHL = RHL.

Exam strategy

When a piecewise function is given, always compute LHL and RHL separately at the junction point. If they differ, the limit does not exist — a favourite JEE trick.

Algebra of Limits and Standard Limits

Sum rule:
$\lim(f + g) = \lim f + \lim g$
.
Difference rule:
$\lim(f - g) = \lim f - \lim g$
.
Product rule:
$\lim(f \cdot g) = \lim f \cdot \lim g$
.
Quotient rule:
$\lim\frac{f}{g} = \frac{\lim f}{\lim g}$
provided
$\lim g \neq 0$
.
Scalar multiple:
$\lim(kf) = k\,\lim f$
.

Trigonometric limit:
$\lim_{x \to 0}\frac{\sin x}{x} = 1$
.
Cosine limit:
$\lim_{x \to 0}\frac{1 - \cos x}{x^2} = \frac{1}{2}$
.
Power limit:
$\lim_{x \to a}\frac{x^n - a^n}{x - a} = n\,a^{n-1}$
for any positive integer n.
Exponential limit:
$\lim_{x \to 0}\frac{e^x - 1}{x} = 1$
.

Degrees vs radians

The limit sinx/x → 1 holds only when x is in radians. In degree mode the result becomes π/180. Always use radian measure for calculus.

Continuity of a Function at a Point

Continuity at x = a

A function f is continuous at x = a if all three conditions hold: (i) f(a) is defined, (ii) $\lim_{x \to a} f(x)$ exists, and (iii) the limit equals the function value: $\lim_{x \to a} f(x) = f(a)$ .

A function is continuous on an interval if it is continuous at every point in that interval.
Polynomials, rational functions, trigonometric functions and exponentials are continuous in their domains.
A jump, hole or vertical asymptote makes the function discontinuous at that point.

Derivative as a Limit (First Principle)

Derivative (first principle)

f'(x) = \lim_{h \to 0}\frac{f(x+h) - f(x)}{h}

The derivative of f at x is the limit of the average rate of change as the interval h shrinks to zero. Geometrically, it gives the slope of the tangent to the curve y = f(x) at the point (x, f(x)).

If the limit does not exist, f is not differentiable at that point.
Differentiability implies continuity, but continuity does not guarantee differentiability (e.g. |x| at x = 0).
The derivative is also written as dy/dx or df/dx.

Algebra of Derivatives

Sum rule:
$(f + g)' = f' + g'$
.
Difference rule:
$(f - g)' = f' - g'$
.
Constant multiple:
$(cf)' = c\,f'$
.
Product rule:
$(uv)' = u'v + uv'$
.
Quotient rule:
$\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$
, provided v ≠ 0.

Chain rule preview

The chain rule (d/dx)[f(g(x))] = f'(g(x))·g'(x) is introduced more formally in Class 12, but many Class 11 problems already require it for composite functions.

Derivatives of Standard Functions

Constant:
$\frac{d}{dx}(c) = 0$
.
Power rule:
$\frac{d}{dx}(x^n) = n\,x^{n-1}$
for any real n.
Trigonometric:
$\frac{d}{dx}(\sin x) = \cos x$
,
$\frac{d}{dx}(\cos x) = -\sin x$
,
$\frac{d}{dx}(\tan x) = \sec^2 x$
.
Exponential and logarithmic:
$\frac{d}{dx}(e^x) = e^x$
,
$\frac{d}{dx}(\ln x) = \frac{1}{x}$
.

\frac{d}{dx}(x^n) = n\,x^{n-1},\quad \frac{d}{dx}(\sin x) = \cos x

Key derivatives (power and trig)

Solved Examples

Example: Find $\lim_{x \to 2}\frac{x^3 - 8}{x - 2}$ .

Solution: Factor the numerator as $(x - 2)(x^2 + 2x + 4)$ . Cancel (x − 2) to get $x^2 + 2x + 4$ . Substituting x = 2 gives 4 + 4 + 4 = 12.

Example: Differentiate $f(x) = x^2\sin x$ using the product rule.

Solution: Let u = x² and v = sin x. Then u′ = 2x and v′ = cos x. By the product rule, f′(x) = 2x sin x + x² cos x.

Revision

Key formulas at a glance

Memorise these before attempting numericals — most exam questions hinge on one of them.

First principle

f'(x) = \lim_{h \to 0}\frac{f(x+h) - f(x)}{h}

Limit sinx/x

\lim_{x \to 0}\frac{\sin x}{x} = 1

Limit (1−cosx)/x²

\lim_{x \to 0}\frac{1 - \cos x}{x^2} = \frac{1}{2}

Power limit

\lim_{x \to a}\frac{x^n - a^n}{x - a} = n\,a^{n-1}

Power rule derivative

\frac{d}{dx}(x^n) = n\,x^{n-1}

Derivative of sinx

\frac{d}{dx}(\sin x) = \cos x

Product rule

(uv)' = u'v + uv'

Exam tips

How this chapter is asked

Where this topic appears in CBSE, JEE Main and NEET papers.

A limit exists only when LHL = RHL at the point.
sinx/x → 1 only in radian measure — degrees give π/180 instead.
Differentiability implies continuity, but not vice versa (|x| at origin).
Use the power limit formula to resolve 0/0 forms quickly.
Product and quotient rules are tested directly in board exams.
Check continuity by verifying f(a), lim f(x), and their equality.
For piecewise functions, always compute LHL and RHL at the break point.
The first principle is a guaranteed 2-mark question in CBSE.

FAQ

Common questions

What is the difference between a limit and a derivative?

A limit describes the value a function approaches at a point; a derivative is a specific type of limit — the limit of the difference quotient — that gives the instantaneous rate of change.

When does a limit not exist?

A limit fails to exist when the left-hand and right-hand limits are different, or when the function oscillates without settling to a value.

Why is sinx/x = 1 only for radians?

The Maclaurin expansion sinx = x − x³/6 + ... assumes radian input. In degrees, sinx ≈ πx/180, so the limit becomes π/180 instead of 1.

Can a function be continuous but not differentiable?

Yes. f(x) = |x| is continuous at x = 0 but not differentiable there because the left and right slopes (−1 and +1) do not match.

Continue learning

Mastering this chapter with live help

Notes help, but doubts clear fastest in a live class. Narayan Gurukul Academy (ClassApna) runs small-batch CBSE, JEE and NEET coaching from our Mohali centre and online — with daily doubt support and mock tests.

Book a Free Demo See plans & pricing

One-on-one guidance available · Live online classes across India