Limits and Continuity — JEE Mains Maths Formulas and Traps

Standard Limits and Algebraic Forms

The standard limits

Standard limits

\lim_{x\to0}\frac{\sin x}x=1,\quad\lim_{x\to0}\frac{1-\cos x}{x^2}=\frac12,\quad\lim_{x\to0}\frac{e^x-1}x=1

Factorise or rationalise

Conjugate

\sqrt p-\sqrt q=\frac{p-q}{\sqrt p+\sqrt q}

Common traps

Match the argument exactly

\frac{\sin3x}x

tends to 3, not 1: the angle and the denominator must be the same before the standard limit applies. Multiply and divide to make them match.

Rationalise the right part

Multiply by the conjugate of the surd that causes the 0, and keep the other factor. Rationalising a denominator that is not zero only adds work.

Series Expansions and Unknown Constants

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Limits by series expansion

Key expansions

\sin x=x-\tfrac{x^3}{6}+\cdots,\quad\cos x=1-\tfrac{x^2}{2}+\cdots,\quad e^x=1+x+\tfrac{x^2}{2}+\cdots

Constants that make a limit finite

Matching coefficients

f(x)=c_0+c_1x+\dots:\ \lim_{x\to0}\frac{f(x)}{x^n}\ \text{finite}\Rightarrow c_0=\dots=c_{n-1}=0

Common traps

Expand far enough

Stop too early and everything cancels, leaving

\frac00

again. Expand each function to the power of

x

in the denominator.

Finite is not the same as zero

The lower coefficients must vanish; the

x^n

coefficient need not. Setting it to zero as well answers a different question.

Exponential Limits (1 to the Power Infinity)

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The 1^∞ formula at a point

1 to the power infinity

\lim f(x)^{g(x)}=e^{\lim g(x)\,(f(x)-1)}\quad(f\to1,\ g\to\infty)

The 1^∞ formula at infinity

The e limit

\lim_{n\to\infty}\left(1+\frac an\right)^{bn}=e^{ab}

Common traps

Is it really 1 to the infinity?

If the base tends to something other than 1, the limit is just (base limit)^(power limit) or 0 or infinity. Apply the formula only when the base tends to 1.

The product can tend to 0

In

\left(1+\frac1{n^2}\right)^n

,

g(f-1)=\frac1n\to0

, so the limit is

e^0=1

. Compute the product; do not assume it gives

e

.

Limits at Infinity and Limits of Sums

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The dominant terms

Leading terms

\lim_{x\to\infty}\frac{a_nx^n+\dots}{b_nx^n+\dots}=\frac{a_n}{b_n}

Sum first, then take the limit

Power sums

\sum_{k=1}^nk^2=\frac{n(n+1)(2n+1)}6

Sums as integrals

Riemann sum

\lim_{n\to\infty}\frac1n\sum_{k=1}^{n}f\!\left(\frac kn\right)=\int_0^1f(x)\,dx

Common traps

Infinity minus infinity is not 0

\sqrt{x^2+x}-x

tends to

\frac12

, not 0. Rationalise before comparing.

Many small terms can add to something

Each term

\frac k{n^2}

tends to 0, but there are

n

of them and their sum tends to

\frac12

. Sum first.

Find the right limits of integration

The limits come from where

\frac kn

starts and ends. If

k

runs from

n

to

3n

, the integral is over

[1,3]

.

Limits via Derivatives and Integrals

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Limits that are derivatives

Derivative as a limit

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

Limits of integrals

Leibniz rule

\frac{d}{dx}\int_a^{u(x)}g(t)\,dt=g\big(u(x)\big)\,u'(x)

Common traps

Check the form before L'Hospital

L'Hospital's rule needs

\frac00

or

\frac\infty\infty

. Applied to a determinate form it gives a wrong answer.

Chain rule on the limit

Differentiating

\int_0^{x^2}g(t)\,dt

gives

g(x^2)\cdot2x

, not

g(x^2)

. The factor

u'(x)

is easy to drop.

Greatest Integer and One-Sided Limits

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The greatest integer function

Greatest integer bounds

t-1<[t]\le t

Left and right limits

Existence of a limit

\lim_{x\to a}f\ \text{exists}\ \Leftarrow\ \lim_{x\to a^-}f=\lim_{x\to a^+}f

Common traps

Values near an integer from above

[\cos x]

near 0 is 0, because

\cos x

is just below 1, not equal to it. Ask on which side of the integer the inside approaches.

sin|x| over x

\frac{\sin|x|}x

tends to 1 from the right and

-1

from the left. A limit that uses

|x|

or

\sqrt{x^2}

needs both sides checked.

Continuity at a Point

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Matching the two sides

Continuity

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

Using continuity

Intermediate value theorem

f(a)\,f(b)<0\ \Rightarrow\ f(c)=0\ \text{for some}\ c\in(a,b)

Common traps

Use the right formula on each side

The left limit uses the piece defined for

x<a

, and the value

f(a)

uses whichever piece includes

a

. Mixing them gives a wrong equation.

No sign change, no guarantee

If

f(a)

and

f(b)

have the same sign, the theorem says nothing: there may be two roots or none.

Counting Discontinuities and Non-Differentiability

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Jumps and corners

Where the greatest integer jumps

[g(x)]\ \text{can break only where}\ g(x)\in\mathbb Z

Compositions

Candidate points

\{x:g\ \text{breaks}\}\cup\{x:g(x)\ \text{is a break of}\ f\}

Functions defined as a limit

Powers in the limit

\lim_{n\to\infty}x^{2n}=\begin{cases}0,&|x|<1\\1,&|x|=1\\\infty,&|x|>1\end{cases}

Common traps

A zero factor can hide a jump

In

x[x]

at 0 the jump of

[x]

is multiplied by 0, so the product is continuous. Test each candidate point; do not just count the integer crossings.

Also where the inside lands on a bad point

Checking only where

g

breaks misses the points where

g(x)

equals a break of

f

. Solve

g(x)=

each bad point of

f

.

Compute the boundary separately

At

x=\pm1

the power

x^{2n}

is exactly 1, giving a third value. Leaving the boundary out misses the break.

Limits and Continuity formulas

Standard Limits and Algebraic Forms

The standard limits

Factorise or rationalise

Common traps

Series Expansions and Unknown Constants

Limits by series expansion

Constants that make a limit finite

Common traps

Exponential Limits (1 to the Power Infinity)

The 1^∞ formula at a point

The 1^∞ formula at infinity

Common traps

Limits at Infinity and Limits of Sums

The dominant terms

Sum first, then take the limit

Sums as integrals

Common traps

Limits via Derivatives and Integrals

Limits that are derivatives

Limits of integrals

Common traps

Greatest Integer and One-Sided Limits

The greatest integer function

Left and right limits

Common traps

Continuity at a Point

Matching the two sides

Using continuity

Common traps

Counting Discontinuities and Non-Differentiability

Jumps and corners

Compositions

Functions defined as a limit

Common traps

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