JEE Maths›Limits, Continuity & Differentiability›Formulas
Formula Sheet

Limits, Continuity & Differentiability Formulas

All key formulas grouped by subtopic. Each one has a quick reminder and common mistakes to watch for.

10 formulas · 7 subtopics

Limit of sin x / x

#1
lim⁡x→0sin⁡xx=1\lim_{x \to 0} \frac{\sin x}{x} = 1

💡 Works for radians only. Also: lim(tan x / x) = 1 and lim(sin(kx) / (kx)) = 1 as x approaches 0.

Exponential Limit (1 + x)^(1/x)

#2
lim⁡x→0(1+x)1/x=eor equivalentlylim⁡n→∞(1+1n)n=e\lim_{x \to 0} (1 + x)^{1/x} = e \quad \text{or equivalently} \quad \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e

💡 For the general form: lim [1 + f(x)]^(1/f(x)) = e when f(x) approaches 0. Use this to handle 1^infinity forms.

Limit of (e^x - 1)/x

#3
lim⁡x→0ex−1x=1\lim_{x \to 0} \frac{e^x - 1}{x} = 1

💡 Useful for converting exponential limits into simpler forms. Substitute t = e^x - 1 when needed.

Limit of (a^x - 1)/x

#4
lim⁡x→0ax−1x=log⁡a(a>0)\lim_{x \to 0} \frac{a^x - 1}{x} = \log a \quad (a > 0)

💡 Here log means log base e (natural logarithm). This follows from writing a^x = e^(x log a) and using the (e^t - 1)/t limit.

Limit of (x^n - a^n)/(x - a)

#5
lim⁡x→axn−anx−a=n⋅an−1\lim_{x \to a} \frac{x^n - a^n}{x - a} = n \cdot a^{n-1}

💡 Valid for all real n. This is essentially the derivative of x^n at x = a from first principles.

L'Hopital's Rule

#6
If lim⁡x→af(x)g(x)=00 or ±∞±∞, then lim⁡x→af(x)g(x)=lim⁡x→af′(x)g′(x)\text{If } \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{0}{0} \text{ or } \frac{\pm\infty}{\pm\infty}, \text{ then } \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}

💡 Always verify the 0/0 or infinity/infinity indeterminate form before applying. May need to be applied repeatedly.

⚠ Applying L'Hopital's rule when the limit is NOT in 0/0 or infinity/infinity form gives wrong answers.

Condition for Continuity

#7
f is continuous at x=a  ⟺  lim⁡x→a−f(x)=lim⁡x→a+f(x)=f(a)f \text{ is continuous at } x = a \iff \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = f(a)

💡 All three must exist and be equal: left-hand limit, right-hand limit, and the function value at the point.

Derivative from First Principles

#8
f′(a)=lim⁡h→0f(a+h)−f(a)hf'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}

💡 Also called the limit definition of derivative. Both left-hand derivative (h approaches 0 from negative side) and right-hand derivative must be equal for differentiability.

Rolle's Theorem

#9
If f is continuous on [a,b], differentiable on (a,b), and f(a)=f(b), then ∃ c∈(a,b) such that f′(c)=0\text{If } f \text{ is continuous on } [a,b], \text{ differentiable on } (a,b), \text{ and } f(a)=f(b), \text{ then } \exists\, c \in (a,b) \text{ such that } f'(c)=0

💡 All three conditions must hold: continuity on [a,b], differentiability on (a,b), and f(a) = f(b). If any fails, the theorem cannot be applied.

Lagrange's Mean Value Theorem (LMVT)

#10
If f is continuous on [a,b] and differentiable on (a,b), then ∃ c∈(a,b) such that f′(c)=f(b)−f(a)b−a\text{If } f \text{ is continuous on } [a,b] \text{ and differentiable on } (a,b), \text{ then } \exists\, c \in (a,b) \text{ such that } f'(c) = \frac{f(b)-f(a)}{b-a}

💡 LMVT is a generalization of Rolle's theorem (Rolle's is the special case when f(a) = f(b)). Geometrically, the tangent at c is parallel to the secant joining (a, f(a)) and (b, f(b)).

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