Limits Continuity and Differentiability formulas for JEE Main & Advanced

24 formula entries for Limits Continuity and Differentiability, in the order the chapter builds them. Each shows the formula, when to use it and the first trap to avoid.

Limit of a Function

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

When to use. When analyzing the behavior of a function near a point, especially for continuity and differentiability checks.

Trap. Assuming $f (a)$ must equal the limit; the limit exists even if $f (a)$ is undefined.

Continuity at a Point

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

When to use. When determining if a function has no jumps, holes, or asymptotes at a point, often for piecewise functions.

Trap. Forgetting to check that $f (a)$ is defined; a hole in the graph means discontinuity.

Differentiability at a Point

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

When to use. When checking if a function has a well-defined tangent or rate of change at a point, especially for piecewise functions.

Trap. Assuming differentiability from continuity; continuity does not guarantee differentiability (e.g., $∣ x ∣$ at $x = 0$ ).

Standard Limits: Trigonometric

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

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

When to use. When evaluating limits involving sine, cosine, or tangent as the variable approaches 0, often after algebraic manipulation.

Trap. Applying $\lim_{x \to 0} \frac{\sin x}{x} = 1$ when the argument is not $x$ (e.g., $\frac{\sin 2 x}{x}$ requires adjustment).

Standard Limits: Exponential and Logarithmic

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

\lim_{x \to 0} \frac{\ln (1 + x)}{x} = 1

\lim_{x \to \infty} (1 + \frac{1}{x})^{x} = e

When to use. When evaluating limits involving exponential or logarithmic functions, especially in indeterminate forms like $0 / 0$ or $1^{\infty}$ .

Trap. Applying $\lim_{x \to 0} \frac{e^{x} - 1}{x} = 1$ when the exponent is not $x$ (e.g., $\frac{e^{2 x} - 1}{x}$ needs factoring).

L'Hôpital's Rule

\lim_{x \to a} \frac{f (x)}{g (x)} = \lim_{x \to a} \frac{f^{'} (x)}{g^{'} (x)}

for

\frac{0}{0}

\frac{\infty}{\infty}

When to use. When direct substitution yields $0 / 0$ or $\infty / \infty$ , especially with trigonometric, exponential, or logarithmic functions.

Trap. Applying L'Hôpital's rule when the limit is not $\frac{0}{0}$ or $\frac{\infty}{\infty}$ ; it gives incorrect results otherwise.

Rationalization for Limits

\lim_{x \to a} \frac{\sqrt{x} - \sqrt{a}}{x - a} = \frac{1}{2 \sqrt{a}}

after rationalization

When to use. When evaluating limits with square roots, cube roots, or other radicals that give $0 / 0$ form.

Trap. Forgetting to multiply both numerator and denominator by the conjugate, leading to an incorrect expression.

Squeeze Theorem (Sandwich Theorem)

g (x) \leq f (x) \leq h (x)

and

\lim g = \lim h = L ⟹ \lim f = L

When to use. When direct evaluation is impossible and the function is bounded between two simpler functions with known limits, e.g., $\lim_{x \to 0} x^{2} \sin (1 / x)$ .

Trap. Using the theorem without verifying the inequality holds for all $x$ near $a$ (except possibly at $a$ ).

Types of Discontinuity

Removable:

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

exists (finite) but

f (a)

is undefined or

f (a) \neq L

; Jump:

\lim_{x \to a^{-}} f (x) \neq \lim_{x \to a^{+}} f (x)

When to use. When analyzing the nature of a break in a function, especially for piecewise or rational functions.

Trap. Calling a discontinuity 'removable' when the limit does not exist; it must exist for removal.

Derivative as a Limit

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

When to use. When deriving derivative formulas from first principles or checking differentiability at a point.

Trap. Using $h \to 0$ but forgetting that $h$ can be negative; the limit must be two-sided.

Chain Rule for Derivatives

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

When to use. When differentiating composite functions, such as $\sin (\ln x)$ , $e^{x^{2}}$ , or $\sqrt{1 + x^{2}}$ .

Trap. Forgetting to multiply by the derivative of the inner function, e.g., differentiating $\sin (x^{2})$ as $\cos (x^{2})$ instead of $2 x \cos (x^{2})$ .

L'Hôpital's Rule for $1^{\infty}$ , $0^{0}$ , $\infty^{0}$ Forms

\lim f (x)^{g (x)} = e^{\lim g (x) \ln f (x)}

When to use. When evaluating limits of the form $f (x)^{g (x)}$ where $f (x) \to 1$ and $g (x) \to \infty$ , or similar.

Trap. Applying L'Hôpital directly to $1^{\infty}$ without taking logs; it's not a $0 / 0$ or $\infty / \infty$ form.

Continuity on an Interval

Continuous on

[a, b]

: continuous at each

x \in (a, b)

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

\lim_{x \to b^{-}} f (x) = f (b)

When to use. When applying theorems like Intermediate Value Theorem or Extreme Value Theorem that require interval continuity.

Trap. Assuming continuity on an open interval implies continuity on the closed interval; endpoints need separate checks.

Intermediate Value Theorem (IVT)

f

continuous on

[a, b]

k

between

f (a)

and

f (b)

⟹ \exists c \in [a, b] : f (c) = k

(and

c \in (a, b)

k

is strictly between)

When to use. When proving existence of roots or solutions to equations, especially for continuous functions.

Trap. Applying IVT when $f$ is not continuous on the closed interval; the theorem fails.

Differentiation from First Principles

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

When to use. When deriving derivative formulas for new functions or checking differentiability at a specific point.

Trap. Forgetting to simplify the numerator before taking the limit; direct substitution gives $0 / 0$ .

Differentiability and Continuity Relationship

f

differentiable at

a ⟹ f

continuous at

a

When to use. When analyzing the smoothness of a function; differentiability ensures no corners or cusps.

Trap. Assuming continuity implies differentiability; counterexample: $f (x) = ∣ x ∣$ at $x = 0$ .

Standard Derivatives

\frac{d}{d x} x^{n} = n x^{n - 1}

\frac{d}{d x} \sin x = \cos x

\frac{d}{d x} e^{x} = e^{x}

\frac{d}{d x} \ln x = \frac{1}{x}

When to use. When differentiating basic functions, especially in combination with product, quotient, and chain rules.

Trap. Forgetting the chain rule when differentiating $\sin (a x)$ ; it's $a \cos (a x)$ , not $\cos (a x)$ .

Implicit Differentiation

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

When to use. When differentiating equations where $y$ cannot be easily expressed as a function of $x$ , e.g., $x^{2} + y^{2} = 1$ .

Trap. Forgetting to multiply by $\frac{d y}{d x}$ when differentiating terms with $y$ (e.g., $\frac{d}{d x} y^{2} = 2 y \frac{d y}{d x}$ ).

Logarithmic Differentiation

\ln y = g (x) \ln f (x) ⟹ \frac{1}{y} \frac{d y}{d x} = g^{'} (x) \ln f (x) + g (x) \frac{f^{'} (x)}{f (x)}

When to use. When differentiating functions like $x^{x}$ , $\frac{(x + 1)^{3} \sqrt{x}}{(x - 1)^{2}}$ , or $\sin x^{\cos x}$ .

Trap. Forgetting to multiply by $y$ at the end to isolate $\frac{d y}{d x}$ .

Mean Value Theorem (MVT)

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

for some

c \in (a, b)

When to use. When proving properties of functions, such as monotonicity or existence of points with specific derivative values.

Trap. Applying MVT when $f$ is not differentiable on $(a, b)$ ; e.g., if there is a corner, MVT may fail.

Rolle's Theorem

f (a) = f (b) ⟹ \exists c \in (a, b) : f^{'} (c) = 0

When to use. When proving that a function has a stationary point between two equal values, often used in root analysis.

Trap. Applying Rolle's theorem when $f (a) \neq f (b)$ ; it's a special case of MVT.

Evaluating Limits Using Series Expansion

\sin x = x - \frac{x^{3}}{6} + O (x^{5})

e^{x} = 1 + x + \frac{x^{2}}{2} + O (x^{3})

When to use. When L'Hôpital's rule leads to repeated differentiation or when the limit involves trigonometric/exponential functions near 0.

Trap. Using too few terms; ensure enough terms to cancel the denominator's leading power.

Handling Absolute Value in Limits and Derivatives

∣ x ∣ = {\begin{cases} x & x \geq 0 \\ - x & x < 0 \end{cases}

When to use. When evaluating limits or derivatives of functions like $∣ x - 1 ∣$ , $∣ \sin x ∣$ , or $∣ x ∣^{3}$ .

Trap. Assuming $∣ x ∣$ is differentiable at $x = 0$ ; it's not, as left and right derivatives differ.

Continuity of Composite Functions

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

When to use. When analyzing continuity of functions like $\sin (1 / x)$ or $e^{\sqrt{x}}$ , especially at boundary points.

Trap. Assuming continuity of $f$ at $g (a)$ is enough; $g$ must also be continuous at $a$ .

Read the full chapter: Limits Continuity and Differentiability for JEE, with what JEE asks and the common traps.