Limit, Continuity and Differentiability Formula Sheet — JEE Main Mathematics

Every key Limit, Continuity and Differentiability formula, definition and theorem for JEE Main Mathematics in one place — with common examiner traps and worked examples. Free to read; blurt from memory, then check your gaps.

Syllabus — topics coveredNTA · 10 sub-topics

Real valued functions
Algebra of functions
Polynomial, rational, trigonometric, logarithmic and exponential functions
Graphs of simple functions
Limits, continuity
Differentiation of sum, difference, product and quotient of functions
Differentiation of trigonometric, inverse trigonometric, logarithmic, exponential, composite and implicit functions
Derivatives of order up to two
Rolle's and Lagrange's Mean Value Theorems
Applications of derivatives: rate of change of quantities, monotonic functions, maxima and minima

Limits — Definition & Evaluation

Limit:

lim_{x \to a} f (x) = L

means f(x) gets arbitrarily close to L as

x \to a

. It

exists if and only if

the left-hand limit equals the right-hand limit (LHL

=

RHL); the value f(a) itself is irrelevant to the limit.

How to evaluate

▸ $Substitute first$ — if you get a finite number, that's the limit.
▸ $\frac{0}{0}$ : factor and cancel, or rationalise a surd.
▸ $\frac{\infty}{\infty}$ (as $x \to \infty$ ): divide top and bottom by the highest power of x.
▸ $\infty - \infty$ or $0 \cdot \infty$ : combine into a single fraction first.

🎯 Exam · The seven indeterminate forms

\frac{0}{0}, \frac{\infty}{\infty}, \infty - \infty, 0 \cdot \infty, 1^{\infty}, 0^{0}, \infty^{0}

. Only these need extra work; for

1^{\infty}, 0^{0}, \infty^{0}

take logarithms first.

⚠️ Watch out · 2/0 is not indeterminate

A form like

\frac{2}{0}

\frac{3}{\infty}

determinate

— it is

\infty

(or undefined) and

0

respectively. Only the seven forms above are genuinely indeterminate.

★ Remember · Use LHL and RHL for piecewise/GIF

For piecewise functions,

∣ x ∣

, the greatest-integer [x] or

\frac{1}{x}

-type behaviour, always compute the LHL (

x \to a^{-}

) and RHL (

x \to a^{+}

) separately — the limit exists only if they match.

🚫 Examiner Trap · Examiner traps

(1) A limit EXISTS only if LHL

=

RHL (both finite and equal) — and that value need NOT equal f(a). (2)

\frac{2}{0}

\frac{3}{0}

is NOT indeterminate — it is

\infty

(or undefined); only

\frac{0}{0}, \frac{\infty}{\infty}, \infty - \infty, 0 \cdot \infty, 1^{\infty}, 0^{0}, \infty^{0}

are. (3) Substitute first; only do extra work if you hit an indeterminate form. (4)

lim (f g) = lim f \cdot lim g

only when both limits EXIST.

Standard Limits & Indeterminate-Form Tricks

Standard limits (as

x \to 0

)

Limit	Value
$\frac{sin x}{x}$	$1$
$\frac{tan x}{x}$	$1$
$\frac{1 - cos x}{x ^{2}}$	$\frac{1}{2}$
$\frac{e ^{x} - 1}{x}$	$1$
$\frac{a ^{x} - 1}{x}$	$ln a$
$\frac{ln ( 1 + x )}{x}$	$1$
$(1 + x)^{1/ x}$	$e$
$\frac{x ^{n} - a ^{n}}{x - a}$ (as $x \to a$ )	$n a^{n - 1}$

Memorise (all as x → 0 unless shown)

▸ $\frac{sin x}{x} \to 1$ , $\frac{tan x}{x} \to 1$ , $\frac{1 - cos x}{x ^{2}} \to \frac{1}{2}$ .
▸ $\frac{e ^{x} - 1}{x} \to 1$ , $\frac{a ^{x} - 1}{x} \to ln a$ , $\frac{ln ( 1 + x )}{x} \to 1$ .
▸ $(1 + x)^{1/ x} \to e$ and $(1 + \frac{1}{n})^{n} \to e$ .
▸ $\frac{x ^{n} - a ^{n}}{x - a} \to n a^{n - 1}$ as $x \to a$ .

★ Remember · L'Hopital's rule

For a

\frac{0}{0}

\frac{\infty}{\infty}

form,

lim \frac{f}{g} = lim \frac{f ^{'}}{g ^{'}}

— differentiate numerator and denominator

separately

(not as a quotient), then re-evaluate; repeat if still indeterminate.

🎯 Exam · The 1^∞ shortcut

f \to 1

and

g \to \infty

, then

lim f^{g} = e^{l i m g (f - 1)}

. For example

lim_{x \to 0} (1 + sin x)^{1/ x} = e^{l i m s i n x / x} = e^{1} = e

⚠️ Watch out · L'Hopital only for 0/0 or ∞/∞

You may only apply L'Hopital once the form is genuinely

\frac{0}{0}

\frac{\infty}{\infty}

. For

1^{\infty}, 0 \cdot \infty, \infty - \infty

, first convert to one of those (take logs or combine fractions).

🚫 Examiner Trap · Examiner traps

(1) The standard limits hold as

x \to 0

in RADIANS:

\frac{s i n x}{x} \to 1

\frac{1 - c o s x}{x ^{2}} \to \frac{1}{2}

(note the

x^{2}

). (2)

1^{\infty}

form:

lim f^{g} = e^{l i m g (f - 1)}

when

f \to 1, g \to \infty

— don't just write

1

. (3) L'Hôpital applies ONLY to

\frac{0}{0}

\frac{\infty}{\infty}

; convert other forms first. (4)

\frac{a ^{x} - 1}{x} \to ln a

(base matters);

(1 + x)^{1/ x} \to e

Continuity

Continuity at a point: f is continuous at

x = a

iff

lim_{x \to a} f (x) = f (a)

— that is,

LHL = RHL = f (a)

, with all three existing and equal. A break in any one of these makes f discontinuous there.

The definition of continuity, sketches of removable, jump and infinite discontinuities, and the properties of continuous functions — Continuity, the kinds of discontinuity, and the properties.

Types of discontinuity

▸ $Removable$ : the limit exists but $\neq = f (a)$ (a single hole).
▸ $Jump$ : LHL and RHL both exist but differ.
▸ $Infinite$ : a one-sided limit is $\pm \infty$ (a vertical asymptote).
▸Polynomials, $sin, cos, e^{x}$ are continuous everywhere.

🎯 Exam · Continuity at a join

For a piecewise f, continuity at the join

x = a

needs

LHL = RHL = f (a)

— set up these equalities to solve for any unknown constants in the definition.

★ Remember · Building continuous functions

Sums, differences, products, quotients (where the denominator is nonzero) and

composites

of continuous functions are continuous. So you rarely check from the definition unless there is a join or a removable hole.

⚠️ Watch out · Limit existing ≠ continuous

A function can have a perfectly good limit at a yet be discontinuous there if f(a) is missing or has the 'wrong' value (a removable discontinuity). Continuity requires the limit to

equal

f(a).

🚫 Examiner Trap · Examiner traps

(1) Continuity at a needs ALL THREE:

lim_{x \to a} f (x)

exists, f(a) defined, and they're EQUAL. (2) Removable (hole) vs jump vs infinite discontinuity — a redefinable point is removable only. (3) At a join of a piecewise function set LHL

=

RHL

= f (a)

to find the unknown constant. (4) A function continuous on a CLOSED interval attains its max & min there (and every value between).

Differentiability

Derivative at a point:

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

, the slope of the tangent at

x = a

. It exists iff the

left-hand derivative equals the right-hand derivative

(and both are finite).

The derivative as a limit, the graph of |x| with its corner showing continuity without differentiability, and the key implication that differentiable implies continuous — The derivative, the $∣ x ∣$ corner, and the key implication.

★ Remember · Differentiable ⇒ continuous (not conversely)

Every differentiable function is continuous, but a continuous function need not be differentiable.

∣ x ∣

is the classic example: continuous everywhere, but its corner at

x = 0

has LHD

= - 1 \neq = + 1 =

RHD.

Where differentiability fails

▸ $Corners$ (like $∣ x ∣$ ): the two one-sided slopes differ.
▸ $Cusps$ and $vertical tangents$ : the slope blows up.
▸Any point of $discontinuity$ (a break can't have a tangent).
▸To test at a join: confirm continuity first, then equate LHD and RHD.

🎯 Exam · The standard test at a point

For f defined piecewise around a: (1) check it is continuous at a; (2) compute LHD

= lim_{h \to 0^{-}} \frac{f ( a + h ) - f ( a )}{h}

and RHD similarly; (3) f is differentiable at a iff they are equal and finite.

⚠️ Watch out · Continuous everywhere ≠ differentiable everywhere

Don't assume a continuous-looking function is differentiable. Absolute values,

max / min

definitions, and the greatest-integer function all create corners or breaks where the derivative fails to exist.

🚫 Examiner Trap · Examiner traps

(1) Differentiable

\Rightarrow

continuous, but NOT conversely (

∣ x ∣

is continuous, not differentiable at

0

). (2) Fails at corners, cusps, vertical tangents and any discontinuity. (3) Differentiable at a needs LHD

=

RHD (both finite) — check continuity FIRST, then equate the one-sided derivatives. (4) 'Differentiable at a?' usually hides a corner/break — don't assume.

Differentiation Rules & Standard Derivatives

Standard derivatives

$f (x)$	$f^{'} (x)$
$x^{n}$	$n x^{n - 1}$
$e^{x}$	$e^{x}$
$a^{x}$	$a^{x} ln a$
$ln x$	$\frac{1}{x}$
$sin x$	$cos x$
$cos x$	$- sin x$
$tan x$	$sec^{2} x$
$sec x$	$sec x tan x$
$sin^{- 1} x$	$\frac{1}{1 - x ^{2}}$
$cos^{- 1} x$	$\frac{- 1}{1 - x ^{2}}$
$tan^{- 1} x$	$\frac{1}{1 + x ^{2}}$
$cot^{- 1} x$	$\frac{- 1}{1 + x ^{2}}$

★ Remember · The three rules

(uv)^{'} = u^{'} v + u v^{'}

;

(\frac{u}{v})^{'} = \frac{u ^{'} v - u v ^{'}}{v ^{2}}

; chain:

\frac{d}{d x} f (g (x)) = f^{'} (g (x)) g^{'} (x)

. The chain rule underlies almost every non-trivial derivative.

Standard derivatives

▸ $x^{n} \to n x^{n - 1}$ ; $e^{x} \to e^{x}$ ; $a^{x} \to a^{x} ln a$ ; $ln x \to \frac{1}{x}$ .
▸ $sin x \to cos x$ ; $cos x \to - sin x$ ; $tan x \to sec^{2} x$ .
▸ $sin^{- 1} x \to \frac{1}{1 - x ^{2}}$ ; $tan^{- 1} x \to \frac{1}{1 + x ^{2}}$ .
▸Inverse co-functions just flip the sign: $cos^{- 1} x \to \frac{- 1}{1 - x ^{2}}$ , etc.

🎯 Exam · Differentiate outside-in

For nested functions like

sin (ln (x^{2} + 1))

, differentiate the outermost layer, then multiply by the derivative of the inside, repeatedly:

cos (ln (x^{2} + 1)) \cdot \frac{1}{x ^{2} + 1} \cdot 2 x

⚠️ Watch out · Don't quotient-rule what you can simplify

Simplify before differentiating where possible (cancel, use log/exponent laws, split a sum). A messy quotient rule is a common source of algebra errors when a quick simplification was available.

🚫 Examiner Trap · Examiner traps

(1) Quotient rule numerator is

u^{'} v - u v^{'}

(in THAT order) over

v^{2}

— sign and order matter. (2) Chain rule: differentiate outside-in and MULTIPLY by the inner derivative (most-missed factor). (3)

\frac{d}{d x} a^{x} = a^{x} ln a

(the

ln a

is essential);

\frac{d}{d x} sin^{- 1} x = \frac{1}{1 - x ^{2}}

\frac{d}{d x} tan^{- 1} x = \frac{1}{1 + x ^{2}}

. (4)

cos^{- 1}

and

cot^{- 1}

derivatives carry a MINUS sign.

Methods of Differentiation

★ Remember · Implicit & parametric

Implicit

: differentiate both sides w.r.t. x treating y as a function (every y brings a

\frac{d y}{d x}

via the chain rule), then solve for

\frac{d y}{d x}

Parametric

\frac{d y}{d x} = \frac{d y / d t}{d x / d t}

🎯 Exam · Logarithmic differentiation

For

y = [f (x)]^{g (x)}

(a variable base AND a variable power) or a big product/quotient, take

ln

of both sides first, then differentiate:

\frac{1}{y} \frac{d y}{d x} = \dots

, and multiply back by y at the end.

More tools

▸ $Higher order$ : $\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (\frac{d y}{d x})$ ; differentiate again for $y^{'''}, \dots$
▸ $Inverse function$ : $\frac{d y}{d x} = \frac{1}{d x / d y}$ .
▸ $Substitution$ : simplify inverse-trig forms (e.g. put $x = tan θ$ ) before differentiating.
▸For a parametric second derivative, differentiate $\frac{d y}{d x}$ w.r.t. t then divide by $\frac{d x}{d t}$ .

⚠️ Watch out · Parametric d²y/dx² isn't (d²y/dt²)/(d²x/dt²)

The second parametric derivative is

\frac{d ^{2} y}{d x ^{2}} = \frac{\frac{d}{d t} ( \frac{d y}{d x} )}{d x / d t}

— you differentiate the

first derivative

again,

not

the second t-derivatives separately.

🚫 Examiner Trap · Examiner traps

(1) Logarithmic differentiation is REQUIRED for

y = [f (x)]^{g (x)}

(variable base AND power) — power/exponent rules alone fail. (2) Implicit: differentiate both sides w.r.t. x, treating y as a function (chain rule), then solve for

\frac{d y}{d x}

. (3) Parametric:

\frac{d y}{d x} = \frac{d y / d t}{d x / d t}

, and

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (\frac{d y}{d x})

— NOT

\frac{d ^{2} y / d t ^{2}}{d ^{2} x / d t ^{2}}

. (4) Simplify inverse-trig by substitution (

x = tan θ

) before differentiating.

Applications of Derivatives

Increasing/decreasing and extrema on a curve, the second-derivative test for maxima/minima, and tangents, normals, rate of change and the mean value theorems — Monotonicity, extrema, tangents/normals and the MVTs.

★ Remember · Increasing / decreasing & extrema

f is increasing where

f^{'} (x) > 0

and decreasing where

f^{'} (x) < 0

. Maxima/minima occur at

critical points

(

f^{'} (x) = 0

or undefined); use

f^{''} < 0 \Rightarrow

max,

f^{''} > 0 \Rightarrow

min, or the sign change of f'.

Tangents, normals & rates

▸Tangent at $(x_{0}, y_{0})$ : slope $m = f^{'} (x_{0})$ ; $normal$ slope $= - \frac{1}{m}$ .
▸Related rates: $\frac{d y}{d t} = \frac{d y}{d x} \cdot \frac{d x}{d t}$ (chain rule).
▸On a closed interval, test endpoints as well as interior critical points.
▸Absolute (global) extremum = the largest/smallest of all these candidate values.

🎯 Exam · The mean value theorems

Rolle

: if f is continuous on [a,b], differentiable on (a,b) and

f (a) = f (b)

, then

f^{'} (c) = 0

for some

c \in (a, b)

Lagrange (MVT)

f^{'} (c) = \frac{f ( b ) - f ( a )}{b - a}

for some

c \in (a, b)

⚠️ Watch out · Check the hypotheses first

Rolle's and Lagrange's theorems require f

continuous on

[a,b]

and differentiable on

(a,b). If f has a corner or break in the interval, the theorem need not apply — verify the hypotheses before quoting the conclusion.

🚫 Examiner Trap · Examiner traps

(1) Critical points are where

f^{'} (x) = 0

OR f' is undefined — don't forget the second kind. (2)

f^{''} < 0 \Rightarrow

max,

f^{''} > 0 \Rightarrow

min; if

f^{''} = 0

use the sign change of f'. (3) On a CLOSED interval also test the ENDPOINTS for the global max/min. (4) Rolle needs

f (a) = f (b)

; MVT gives

f^{'} (c) = \frac{f ( b ) - f ( a )}{b - a}

— both need continuity on [a,b] and differentiability on (a,b).

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What are the most important Limit, Continuity and Differentiability formulas for JEE Main?

This Limit, Continuity and Differentiability formula sheet covers all the high-yield Mathematics formulas, definitions and theorems you need for JEE Main, across Real valued functions, Algebra of functions, Polynomial, rational, trigonometric, logarithmic and exponential functions, Graphs of simple functions, Limits, continuity — each shown with the key result and, where useful, a worked example.

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