JEE Mains Maths · Limits and Continuity

Greatest Integer and One-Sided Limits

Limits involving the greatest integer function, the modulus or expressions that behave differently on each side of a point, found from the left and right separately.

All Limits and Continuity notes JEE Mains Maths strategy

Why this matters

Ten PYQs, seven of them multiple choice. Six involve the greatest integer function, often near an integer or with a squeeze; four compare the left and right limits of a modulus or an exponential. Two ideas cover the page.

Concept 1 of 2: The greatest integer function

[x]

is constant between integers and jumps at each integer. Near an integer

n

, take the two sides:

[x]=n-1

just to the left and

n

just to the right. For expressions like

x\left[\frac1x\right]

, squeeze with

t-1<[t]\le t

Definition

$[x]=n$ for $n\le x<n+1$ .
At an integer $n$ : left limit $n-1$ , right limit $n$ .
Squeeze: $t-1<[t]\le t$ ; $\{t\}=t-[t]\in[0,1)$ .

Greatest integer bounds

t-1<[t]\le t

Worked example

Does

\lim_{x\to2}[x]

exist?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2025 · 29 Jan 2025 · Q73Moderate

Example 1 · Limits and Continuity · Greatest Integer and One-Sided Limits

Let

\lbrack t\rbrack

be the greatest integer less than or equal to

t

. Then the least value of

p \in N

for which

\underset{x \rightarrow0^{+}}{\lim} \left( x\left( \left\lbrack \frac{1}{x} \right\rbrack+\left\lbrack \frac{2}{x} \right\rbrack+ \ldots.. +\left\lbrack \frac{p}{x} \right\rbrack \right)-x^{2}\left( \left\lbrack \frac{1}{x^{2}} \right\rbrack+\left\lbrack \frac{2^{2}}{x^{2}} \right\rbrack+ \ldots. +\left\lbrack \frac{9^{2}}{x^{2}} \right\rbrack \right) \right)\geq 1

is equal to

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.

Drill 5 more on the greatest integer function

Concept 2 of 2: Left and right limits

A limit exists only if the left and right limits agree. Moduli,

e^{1/x}

and

\tan^{-1}\frac1x

all behave differently on the two sides of 0, so compute each side separately: for

x<0

|x|=-x

and

e^{1/x}\to0

; for

x>0

e^{1/x}\to\infty

Definition

The limit exists exactly when the left and right limits are equal.
$e^{1/x}\to0$ as $x\to0^-$ , $\to\infty$ as $x\to0^+$ .
$\tan^{-1}\frac1x\to\pm\frac\pi2$ as $x\to0^\pm$ .

Existence of a limit

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

Worked example

Does

\lim_{x\to0}\frac{|x|}x

exist?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2024 · 1 February 2024 · Q85Moderate

Example 2 · Limits and Continuity · Greatest Integer and One-Sided Limits

Let

\{ x\}

denote the fractional part of

x

and

f(x) =\frac{\cos^{- 1}\left( 1 - \{ x\}^{2} \right)\sin^{- 1}\left( 1 - \{ x\} \right)}{\{ x\} - \{ x\}^{3}},x\neq 0

. If

L

and

R

respectively denotes the left hand limit and the right hand limit of

f(x)

x= 0

, then

\frac{32}{\pi^{2}}\left( L^{2}+R^{2} \right)

is equal to

sin|x| over x

\frac{\sin|x|}x

tends to 1 from the right and

-1

from the left. A limit that uses

|x|

\sqrt{x^2}

needs both sides checked.

Drill 3 more on left and right limits

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

The greatest integer function
Greatest integer bounds
$t-1<[t]\le t$
Left and right limits
Existence of a limit
$\lim_{x\to a}f\ \text{exists}\ \Leftarrow\ \lim_{x\to a^-}f=\lim_{x\to a^+}f$

Watch out for (2)

Values near an integer from above
→ The greatest integer function
sin|x| over x
→ Left and right limits

Test yourself on Limits and Continuity

20 past JEE Mains questions from this chapter, timed at 48 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Limits and Continuity chapter test Drill all 10 questions