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Limit formulas

Master Limit through 34 JEE Advanced-level formulas, systematically structured with every variable spelled out. Revise concept-wise, identify the areas where you need improvement, and focus your preparation with greater precision.

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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
$x \to a lim f (x) = l \Leftrightarrow x \to a^{-} lim f = x \to a^{+} lim f = l$
Q1MCQExistence
$\lim_{x\to a}f(x)$ exists iff:
- A
  left limit $=$ right limit
- B
  $f(a)$ is defined
- C
  $f$ is continuous
- D
  $f$ is differentiable
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
Q1MCQ(aˣ-1)/x
$\displaystyle\lim_{x\to0}\dfrac{a^{x}-1}{x}$ equals:
- A
  $\log_e a$
- B
  $a$
- C
  $1$
- D
  $\log_a e$
Standard: log(1+x)/x
$x \to 0 lim \frac{lo g _{e} ( 1 + x )}{x} = 1$
Standard: (1-cos x)/x²
$x \to 0 lim \frac{1 - cos x}{x ^{2}} = \frac{1}{2}$
Q1MCQ(1-cos x)/x²
$\displaystyle\lim_{x\to0}\dfrac{1-\cos x}{x^{2}}$ equals:
- A
  $\dfrac12$
- B
  $1$
- C
  $0$
- D
  $2$
Euler: (1+x)^{1/x}
$x \to 0 lim (1 + x)^{1/ x} = e$
Q1MCQ(1+x)^{1/x}
$\displaystyle\lim_{x\to0}(1+x)^{1/x}$ equals:
- A
  $e$
- B
  $1$
- C
  $0$
- D
  $e^{2}$
Euler: (1+1/x)ˣ
$x \to \infty lim (1 + \frac{1}{x})^{x} = e$
Q1MCQ(1+1/x)ˣ
$\displaystyle\lim_{x\to\infty}\left(1+\dfrac1x\right)^{x}$ equals:
- A
  $e$
- B
  $1$
- C
  $\infty$
- D
  $0$
Euler: (1+a/x)ˣ
$x \to \infty lim (1 + \frac{a}{x})^{x} = e^{a}$
Q1MCQ(1+a/x)ˣ
$\displaystyle\lim_{x\to\infty}\left(1+\dfrac{3}{x}\right)^{x}$ equals:
- A
  $e^{3}$
- B
  $e$
- C
  $3e$
- D
  $1$
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)
$x \to \infty lim \frac{f ( x )}{g ( x )} = \frac{lead ( f )}{lead ( g )}$
Q1MCQPolynomials m=n
$\displaystyle\lim_{x\to\infty}\dfrac{2x^{2}+1}{3x^{2}+5}$ equals:
- A
  $\dfrac23$
- B
  $0$
- C
  $\infty$
- D
  $1$
|x-a|/(x-a) at a
$x \to a lim \frac{∣ x - a ∣}{x - a} does not exist$
Q1MCQ|x-a| limit
$\displaystyle\lim_{x\to0}\dfrac{|x|}{x}$:
- A
  does not exist
- B
  $=1$
- C
  $=-1$
- D
  $=0$
Limit of xⁿ (|x|<1)
$n \to \infty lim x^{n} = 0, ∣ x ∣ < 1$
L'Hospital's rule
$lim \frac{f}{g} = lim \frac{f ^{'}}{g ^{'}} (\frac{0}{0} or \frac{\infty}{\infty})$
Q1MCQL'Hospital
L'Hospital's rule applies to a limit of the form:
- A
  $\dfrac00$ or $\dfrac{\infty}{\infty}$
- B
  $1+1$
- C
  any limit
- D
  $2\cdot3$
1^∞ form
$lim f^{g} = e^{l i m g (f - 1)}, f \to 1, g \to \infty$
Q1MCQ1^∞ form
For $f\to1,\ g\to\infty$, $\lim f^{g}$ equals:
- A
  $e^{\lim g(f-1)}$
- B
  $1$
- C
  $\lim f\cdot\lim g$
- D
  $e^{\lim (f-1)}$
0·∞ / conversion
$0 \cdot \infty \to \frac{0}{0} or \frac{\infty}{\infty}$
rewrite before applying L'Hospital
Q1MCQ0·∞
A $0\cdot\infty$ form is handled by:
- A
  rewriting it as $\tfrac00$ or $\tfrac{\infty}{\infty}$
- B
  setting it to $0$
- C
  setting it to $\infty$
- D
  setting it to $1$
Indeterminate forms
$\frac{0}{0}, \frac{\infty}{\infty}, \infty - \infty, 0 \cdot \infty, 0^{0}, \infty^{0}, 1^{\infty}$
Q1MCQIndeterminate
Which of these is an indeterminate form?
- A
  $\dfrac00$
- B
  $\dfrac10$
- C
  $0^{1}$
- D
  $1^{0}$
Sandwich theorem
$f \leq g \leq h, lim f = lim h = l \Rightarrow lim g = l$
Q1MCQSandwich
If $f\le g\le h$ and $\lim f=\lim h=5$, then $\lim g$ is:
- A
  $5$
- B
  $0$
- C
  undetermined
- D
  $10$
x·sin(1/x)
$x \to 0 lim x sin \frac{1}{x} = 0$
bounded × infinitesimal
sin(1/x) at 0
$x \to 0 lim sin \frac{1}{x} does not exist$
Q1MCQsin(1/x)
$\displaystyle\lim_{x\to0}\sin\dfrac1x$:
- A
  does not exist
- B
  $=0$
- C
  $=1$
- D
  $=\infty$
eˣ series
$e^{x} = 1 + \frac{x}{1 !} + \frac{x ^{2}}{2 !} + \dots$
Q1MCQeˣ series
The coefficient of $x^{2}$ in the expansion of $e^{x}$ is:
- A
  $\dfrac12$
- B
  $1$
- C
  $\dfrac16$
- D
  $2$
sin x series
$sin x = x - \frac{x ^{3}}{3 !} + \frac{x ^{5}}{5 !} - \dots$
Q1MCQsin x series
The first two terms of the expansion of $\sin x$ are:
- A
  $x-\dfrac{x^{3}}{6}$
- B
  $1-\dfrac{x^{2}}{2}$
- C
  $x+\dfrac{x^{3}}{6}$
- D
  $x-\dfrac{x^{2}}{2}$
cos x series
$cos x = 1 - \frac{x ^{2}}{2 !} + \frac{x ^{4}}{4 !} - \dots$
Q1MCQcos x series
The expansion of $\cos x$ begins:
- A
  $1-\dfrac{x^{2}}{2}+\cdots$
- B
  $x-\dfrac{x^{3}}{6}+\cdots$
- C
  $1+\dfrac{x^{2}}{2}+\cdots$
- D
  $1-x+\cdots$
log(1+x) series
$lo g_{e} (1 + x) = x - \frac{x ^{2}}{2} + \frac{x ^{3}}{3} - \dots, ∣ x ∣ < 1$
Q1MCQlog(1+x) series
The expansion of $\log_e(1+x)$ begins:
- A
  $x-\dfrac{x^{2}}{2}+\dfrac{x^{3}}{3}-\cdots$
- B
  $1+x+x^{2}+\cdots$
- C
  $x+\dfrac{x^{2}}{2}+\cdots$
- D
  $-x+\dfrac{x^{2}}{2}-\cdots$
(1+x)^p series
$(1 + x)^{p} = 1 + p x + \frac{p ( p - 1 )}{2 !} x^{2} + \dots, ∣ x ∣ < 1$
Q1MCQ(1+x)^p series
The coefficient of $x$ in $(1+x)^{p}$ is:
- A
  $p$
- B
  $1$
- C
  $\dfrac{p(p-1)}{2}$
- D
  $p^{2}$

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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

More JEE Advanced Mathematics formula sheets

Other ways to revise this chapter

Preparing students for India’s top institutes

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