CalculusVery High Yield (3-4 Qs in JEE)

Limits, Continuity & Differentiability Formulas

Fundamental algebraic, exponential, logarithmic, and trigonometric limits, continuity conditions, and derivative theorems.

9 Formulas•868+ Questions in Bank•

Concept Diagram

Definite Integral & Tangent Line

Geometric interpretation of ∫ f(x) dx as area under curve and derivative f'(x₀) as tangent slope.

Core Formulas & Mathematical Identities

Standard Algebraic & Trig Limits

\lim_{x \to 0} \frac{\sin x}{x} = 1, \quad \lim_{x \to 0} \frac{\tan x}{x} = 1, \quad \lim_{x \to a} \frac{x^n - a^n}{x - a} = n a^{n-1}

Fundamental trigonometric and polynomial limits for resolving 0/0 indeterminate forms.

Examiner Tip: Angle x MUST be measured in radians.

Exponential & Logarithmic Limits

\lim_{x \to 0} \frac{e^x - 1}{x} = 1, \quad \lim_{x \to 0} \frac{a^x - 1}{x} = \ln a, \quad \lim_{x \to 0} \frac{\ln(1 + x)}{x} = 1

Standard limits for natural exponential and logarithmic expressions.

Examiner Tip: Useful substitution: lim_{x->0} (ln(1+ax))/x = a.

Exponential 1^∞ Indeterminate Form

\lim_{x \to a} [f(x)]^{g(x)} = e^{\lim_{x \to a} g(x)[f(x) - 1]} \quad (\text{when } f(x) \to 1, g(x) \to \infty)

Converts 1^∞ indeterminate power limits into standard exponential products directly.

Examiner Tip: Very frequent in JEE Main: lim_{x->0} (1 + ax)^(b/x) = e^(ab).

Maclaurin Series Expansions

\sin x = x - \frac{x^3}{6} + \dots, \quad e^x = 1 + x + \frac{x^2}{2} + \dots, \quad \ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \dots

Taylor-Maclaurin series for resolving complex indeterminate forms without repetitive L'Hôpital differentiation.

Examiner Tip: Keep terms up to the lowest non-vanishing power of x in the denominator.

Continuity Condition at a Point

\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = f(c) \iff \text{LHL} = \text{RHL} = f(c)

A function f(x) is continuous at x = c if and only if both one-sided limits exist and equal the functional value.

Examiner Tip: Greatest integer [x], fractional part {x}, and signum sgn(x) are classic discontinuity test cases.

Differentiability from First Principles

f'(c) = \lim_{h \to 0} \frac{f(c + h) - f(c)}{h}, \quad \text{LHD} = \text{RHD} \implies \text{Differentiable}

Differentiability implies continuity, but continuity does NOT imply differentiability.

Examiner Tip: Sharp vertices, cusps, and vertical tangents signify non-differentiable points.

L'Hôpital's Rule for Indeterminate Forms

\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} \quad \left(\text{Valid strictly for } \frac{0}{0} \text{ and } \frac{\infty}{\infty} \text{ forms}\right)

Differentiate numerator and denominator independently until the indeterminate form is resolved.

Examiner Tip: Do NOT apply the quotient rule when applying L'Hôpital's rule. Check that the indeterminate form persists before differentiating repeatedly.

Sandwich (Squeeze) Theorem & Limits

g(x) \le f(x) \le h(x) \quad \text{and} \quad \lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L \implies \lim_{x \to a} f(x) = L

Bounding technique for evaluating limits of functions involving oscillatory factors like [x], {x}, or sin(1/x).

Examiner Tip: Classic example: lim_{x->0} x² sin(1/x) = 0 because -x² ≤ x² sin(1/x) ≤ x².

Intermediate Value Theorem (IVT)

f(x) \in C[a, b] \text{ and } f(a) \cdot f(b) < 0 \implies \exists c \in (a, b) \text{ such that } f(c) = 0

Fundamental root existence theorem for continuous real-valued functions across opposite-sign endpoints.

Examiner Tip: Every odd degree real polynomial has at least one real root.

Top Examiner Traps & Pitfalls

Frequently tested misconceptions in Limits, Continuity & Differentiability

•Applying L'Hôpital's Rule when the limit is NOT an indeterminate form 0/0 or ∞/∞.
•Forgetting to verify both Left-Hand Limit (LHL) and Right-Hand Limit (RHL) at non-differentiable points (like |x| or [x]).
•Forgetting that trigonometric limits require x to be in radians: lim_{x->0} sin(x°)/x = π/180.

Next Chapter

Functions & Relations

Calculus•8 Formulas