Limit formulas

Every formula JEE Advanced expects you to know for Limit, with the variables spelled out. Free, no sign-up. Read it before a mock, not during one.

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Formulas: 34
Subject: Mathematics
Exam: JEE Advanced
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Limit, every formula

34 formulas, typeset and free. Print it, or keep it open beside your practice.

Existence of a limit
$x \to a lim f (x) = l \Leftrightarrow x \to a^{-} lim f = x \to a^{+} lim f = l$
Sum/difference rule
$lim (f \pm g) = lim f \pm lim g$
Product rule
$lim (f \cdot g) = (lim f) (lim g)$
Quotient rule
$lim \frac{f}{g} = \frac{lim f}{lim g}, lim g \neq = 0$
Power rule
$lim [f (x)]^{k} = l^{k}$
Standard: (xⁿ-aⁿ)/(x-a)
$x \to a lim \frac{x ^{n} - a ^{n}}{x - a} = n a^{n - 1}$
Standard: sin x / x
$x \to 0 lim \frac{sin x}{x} = 1$
x in radians
Standard: tan x / x
$x \to 0 lim \frac{tan x}{x} = 1$
Standard: sin ax / x
$x \to 0 lim \frac{sin a x}{x} = a$
Standard: arcsin x / x
$x \to 0 lim \frac{sin ^{- 1} x}{x} = 1$
Standard: arctan x / x
$x \to 0 lim \frac{tan ^{- 1} x}{x} = 1$
Standard: (eˣ-1)/x
$x \to 0 lim \frac{e ^{x} - 1}{x} = 1$
Standard: (aˣ-1)/x
$x \to 0 lim \frac{a ^{x} - 1}{x} = lo g_{e} a$
a>0
Standard: log(1+x)/x
$x \to 0 lim \frac{lo g _{e} ( 1 + x )}{x} = 1$
Standard: (1-cos x)/x²
$x \to 0 lim \frac{1 - cos x}{x ^{2}} = \frac{1}{2}$
Euler: (1+x)^{1/x}
$x \to 0 lim (1 + x)^{1/ x} = e$
Euler: (1+1/x)ˣ
$x \to \infty lim (1 + \frac{1}{x})^{x} = e$
Euler: (1+a/x)ˣ
$x \to \infty lim (1 + \frac{a}{x})^{x} = e^{a}$
Ratio of polynomials (m<n)
$x \to \infty lim \frac{f ( x )}{g ( x )} = 0, de g f < de g g$
Ratio of polynomials (m=n)
$x \to \infty lim \frac{f ( x )}{g ( x )} = \frac{lead ( f )}{lead ( g )}$
|x-a|/(x-a) at a
$x \to a lim \frac{∣ x - a ∣}{x - a} does not exist$
Limit of xⁿ (|x|<1)
$n \to \infty lim x^{n} = 0, ∣ x ∣ < 1$
L'Hospital's rule
$lim \frac{f}{g} = lim \frac{f ^{'}}{g ^{'}} (\frac{0}{0} or \frac{\infty}{\infty})$
1^∞ form
$lim f^{g} = e^{l i m g (f - 1)}, f \to 1, g \to \infty$
0·∞ / conversion
$0 \cdot \infty \to \frac{0}{0} or \frac{\infty}{\infty}$
rewrite before applying L'Hospital
Indeterminate forms
$\frac{0}{0}, \frac{\infty}{\infty}, \infty - \infty, 0 \cdot \infty, 0^{0}, \infty^{0}, 1^{\infty}$
Sandwich theorem
$f \leq g \leq h, lim f = lim h = l \Rightarrow lim g = l$
x·sin(1/x)
$x \to 0 lim x sin \frac{1}{x} = 0$
bounded × infinitesimal
sin(1/x) at 0
$x \to 0 lim sin \frac{1}{x} does not exist$
eˣ series
$e^{x} = 1 + \frac{x}{1 !} + \frac{x ^{2}}{2 !} + \dots$
sin x series
$sin x = x - \frac{x ^{3}}{3 !} + \frac{x ^{5}}{5 !} - \dots$
cos x series
$cos x = 1 - \frac{x ^{2}}{2 !} + \frac{x ^{4}}{4 !} - \dots$
log(1+x) series
$lo g_{e} (1 + x) = x - \frac{x ^{2}}{2} + \frac{x ^{3}}{3} - \dots, ∣ x ∣ < 1$
(1+x)^p series
$(1 + x)^{p} = 1 + p x + \frac{p ( p - 1 )}{2 !} x^{2} + \dots, ∣ x ∣ < 1$

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Limit formulas

Limit, every formula

Existence of a limit

Sum/difference rule

Product rule

Quotient rule

Power rule

Standard: (xⁿ-aⁿ)/(x-a)

Standard: sin x / x

Standard: tan x / x

Standard: sin ax / x

Standard: arcsin x / x

Standard: arctan x / x

Standard: (eˣ-1)/x

Standard: (aˣ-1)/x

Standard: log(1+x)/x

Standard: (1-cos x)/x²

Euler: (1+x)^{1/x}

Euler: (1+1/x)ˣ

Euler: (1+a/x)ˣ

Ratio of polynomials (m<n)

Ratio of polynomials (m=n)

|x-a|/(x-a) at a

Limit of xⁿ (|x|<1)

L'Hospital's rule

1^∞ form

0·∞ / conversion

Indeterminate forms

Sandwich theorem

x·sin(1/x)

sin(1/x) at 0

eˣ series

sin x series

cos x series

log(1+x) series

(1+x)^p series

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Know the Limit formulas. Now use them.

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