When does a limit fail to exist?

A two-sided limit fails to exist if the left-hand and right-hand limits are unequal, if at least one one-sided limit is infinite, or if the function oscillates without settling near the point.

Why is 0/0 called indeterminate instead of undefined?

Undefined means there is no answer. Indeterminate means the form alone does not determine the value; different functions of type 0/0 can give different limits, so more work is needed.

Do I need the epsilon-delta definition for JEE?

JEE problems are almost always solved by intuition plus algebra, standard limits, expansions, or L'Hospital's rule. The formal definition is rarely tested directly but helps understanding.

When should I compute LHL and RHL separately?

Whenever the function's expression changes at the point. Common triggers: piecewise functions, absolute values, the greatest integer function, the fractional part, and step or sign functions.

Can a limit exist if the function is not defined at the point?

Yes. For example, (x squared minus 1) divided by (x minus 1) is undefined at x equals 1 but its limit at 1 is 2, because the expression simplifies to x plus 1 for all other x.

Introduction To Limits

Q: What is the difference between the limit of f(x) as x approaches a and f(a)?

The limit is the value f(x) approaches as x gets close to a (but x is not equal to a). f(a) is the actual value of the function at a. They may be different, and f(a) may not exist at all while the limit still exists. When they agree, the function is continuous at a.

Q: Is the limit at infinity a real number?

If it exists as a finite value, yes. If we write limit equals infinity, we mean the limit does not exist as a finite real number and the function grows without bound.

Q: What is the relationship between limits and continuity?

A function is continuous at a point if and only if the left-hand limit, right-hand limit, and function value all exist and agree. Continuity is limit existence plus agreement with the function's value.

MathsLimits, Continuity And DifferentiabilityFor JEE aspirants

The limit of a function $f (x)$ as $x$ approaches $a$ is the value $L$ that $f (x)$ gets arbitrarily close to as $x$ gets arbitrarily close to $a$ , written $lim_{x \to a} f (x) = L$ . The limit exists only when the left-hand limit (LHL) and right-hand limit (RHL) both exist and are equal to the same finite value. Crucially, the limit describes the approach, not the value $f (a)$ itself, so a limit can exist even where the function is undefined. Limits are the foundation of continuity, derivatives, and integrals in JEE calculus.

Key Formulas - Quick Reference

Limit: $lim_{x \to a} f (x) = L$ if $f (x) \to L$ as $x \to a$ (from both sides).
Left-hand limit: $lim_{x \to a^{-}} f (x) = lim_{h \to 0^{+}} f (a - h)$ .
Right-hand limit: $lim_{x \to a^{+}} f (x) = lim_{h \to 0^{+}} f (a + h)$ .
Existence: $lim_{x \to a} f (x)$ exists $⟺$ LHL $=$ RHL $=$ finite.
The seven indeterminate forms: $\frac{0}{0}, \frac{\infty}{\infty}, 0 \times \infty, 0^{0}, \infty^{0}, \infty - \infty, 1^{\infty}$ .
Algebra of limits: if $lim f = ℓ$ and $lim g = m$ (both finite), then $lim (f \pm g) = ℓ \pm m$ , $lim (f g) = ℓ m$ , $lim (f / g) = ℓ / m$ if $m \neq = 0$ .
$ε$ - $δ$ definition: $lim_{x \to a} f (x) = L$ iff for every $ε > 0$ there exists $δ > 0$ such that $0 < ∣ x - a ∣ < δ ⟹ ∣ f (x) - L ∣ < ε$ .

1. What is a Limit? The Intuitive Idea

Consider the function $f (x) = \frac{sin x}{x}$ . If we substitute $x = 0$ , we get $\frac{0}{0}$ , which is undefined. Yet as $x$ takes values closer and closer to $0$ (like $0.1, 0.01, 0.001, \dots$ ), the value of $f (x)$ gets closer and closer to $1$ . We write this observation as

x \to 0 lim \frac{sin x}{x} = 1.

The key idea: a limit describes the behaviour of the function near a point, not the value at the point. In fact, the value $f (a)$ is completely irrelevant to computing $lim_{x \to a} f (x)$ ; the limit only cares about $x$ values near $a$ , but not $x = a$ itself.

Figure 1: As

x

approaches

a

from either side,

f (x)

approaches the limit value

L

. The open circle indicates that

f (a)

itself may be undefined or different from

L

2. Formal Definition of a Limit

2.1 Informal (working) definition

We say

lim_{x \to a} f (x) = L

if we can make

f (x)

as close to

L

as we want, for all

x

sufficiently close to

a

from both sides, without actually letting

x

equal

a

2.2 Formal $ε$ - $δ$ definition

Let $f (x)$ be defined on an interval containing $x = a$ , except possibly at $x = a$ itself. Then

lim_{x \to a} f (x) = L

if and only if for every number

ε > 0

, there exists a number

δ > 0

such that

∣ f (x) - L ∣ < ε whenever 0 < ∣ x - a ∣ < δ .

Note the strict inequality $0 < ∣ x - a ∣$ : this is what enforces $x \neq = a$ . The limit process explicitly excludes the point $x = a$ from consideration.

Symbolic reminder: writing

x \to a

under a limit already implies

x \neq = a

. That is why

lim_{x \to a} f (x)

can exist and equal

L

even when

f (a)

is undefined or when

f (a) \neq = L

3. One-Sided Limits: LHL and RHL

Since $x$ can approach $a$ from two directions on the real line, we define two one-sided limits.

3.1 Right-Hand Limit (RHL)

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

f (x)

can be made as close to

L

as we want for all

x

sufficiently close to

a

with

x > a

A common substitution for computation is to write $x = a + h$ where $h \to 0^{+}$ :

x \to a^{+} lim f (x) = h \to 0^{+} lim f (a + h) .

Solved Example 1

Evaluate the right-hand limit of

f (x) = ⎩ ⎨ ⎧ \frac{∣ x - 4∣}{x - 4}, x \neq = 4 0, x = 4

x = 4

Solution:

Setting $x = 4 + h$ with $h \to 0^{+}$ :

x \to 4^{+} lim f (x) = h \to 0^{+} lim \frac{∣4 + h - 4∣}{4 + h - 4} = h \to 0^{+} lim \frac{∣ h ∣}{h} .

Since $h \to 0^{+}$ means $h > 0$ , we have $∣ h ∣ = h$ , so

h \to 0^{+} lim \frac{h}{h} = h \to 0^{+} lim 1 = 1.

3.2 Left-Hand Limit (LHL)

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

f (x)

can be made as close to

L

as we want for all

x

sufficiently close to

a

with

x < a

Substitute $x = a - h$ where $h \to 0^{+}$ :

x \to a^{-} lim f (x) = h \to 0^{+} lim f (a - h) .

Solved Example 2

Evaluate the left-hand limit of

f (x) = ⎩ ⎨ ⎧ \frac{∣ x - 4∣}{x - 4}, x \neq = 4 0, x = 4

x = 4

Solution:

Setting $x = 4 - h$ with $h \to 0^{+}$ :

x \to 4^{-} lim f (x) = h \to 0^{+} lim \frac{∣4 - h - 4∣}{4 - h - 4} = h \to 0^{+} lim \frac{∣ - h ∣}{- h} = h \to 0^{+} lim \frac{h}{- h} = - 1.

Figure 2: Graph of

f (x) = \frac{∣ x - 4∣}{x - 4}

near

x = 4

. Since LHL

= - 1 \neq = 1 =

RHL, the two-sided limit

lim_{x \to 4} f (x)

does not exist.

4. Existence of a Limit

Combining the two one-sided limits gives the criterion for existence of the two-sided limit.

Existence criterion.

lim_{x \to a} f (x)

exists and equals

ℓ

if and only if

x \to a^{-} lim f (x) = x \to a^{+} lim f (x) = ℓ (a finite quantity).

If both one-sided limits exist but are unequal, or if either one-sided limit is infinite or fails to exist, the two-sided limit does not exist.

Boundary case: if only one side of

a

lies in the domain of

f

(say only

x > a

makes sense), then

lim_{x \to a} f (x)

is taken to mean the one available one-sided limit, that is,

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

Solved Example 3

f (x) = ⎩ ⎨ ⎧ \frac{x - ∣ x ∣}{x}, x \neq = 0 2, x = 0

, check whether

lim_{x \to 0} f (x)

exists.

Solution:

LHL at $x = 0$ : Put $x = - h$ with $h \to 0^{+}$ .

x \to 0^{-} lim f (x) = h \to 0^{+} lim \frac{- h - ∣ - h ∣}{- h} = h \to 0^{+} lim \frac{- h - h}{- h} = h \to 0^{+} lim \frac{- 2 h}{- h} = 2.

RHL at $x = 0$ : Put $x = h$ with $h \to 0^{+}$ .

x \to 0^{+} lim f (x) = h \to 0^{+} lim \frac{h - ∣ h ∣}{h} = h \to 0^{+} lim \frac{h - h}{h} = h \to 0^{+} lim \frac{0}{h} = 0.

Since LHL $= 2 \neq = 0 =$ RHL, $lim_{x \to 0} f (x)$ does not exist. Notice that $f (0) = 2$ is irrelevant to this conclusion.

5. Indeterminate Forms

When we try to evaluate a limit by direct substitution and the result has no definite value (it depends on how the parts approach their targets), we call it an indeterminate form. There are seven standard indeterminate forms:

Form	Typical source
$\frac{0}{0}$	$lim_{x \to 0} \frac{sin x}{x}$
$\frac{\infty}{\infty}$	$lim_{x \to \infty} \frac{x ^{2} + 1}{2 x ^{2} - 3}$
$0 \times \infty$	$lim_{x \to 0^{+}} x ln x$
$\infty - \infty$	$lim_{x \to \infty} (x^{2} + x - x)$
$0^{0}$	$lim_{x \to 0^{+}} x^{x}$
$\infty^{0}$	$lim_{x \to \infty} x^{1/ x}$
$1^{\infty}$	$lim_{x \to \infty} (1 + \frac{1}{x})^{x}$

Here

0

does not stand for the numerical value

0

but for a quantity approaching

0

; similarly

1

(in

1^{\infty}

) means a quantity approaching

1

, not the constant

1

. This is why

1^{\infty}

is indeterminate even though

1

raised to any finite power is

1

Every indeterminate form has a technique to resolve it, covered in the next concept (Methods to Evaluate Limits) and in L'Hospital's Rule.

6. Algebra of Limits

If $lim_{x \to a} f (x) = ℓ$ and $lim_{x \to a} g (x) = m$ both exist and are finite, then the following hold.

Algebra Rules

Constant multiple: $lim_{x \to a} k f (x) = k ℓ$ for any constant $k$ .
Sum/difference: $lim_{x \to a} (f (x) \pm g (x)) = ℓ \pm m$ .
Product: $lim_{x \to a} f (x) g (x) = ℓ \cdot m$ .
Quotient: $lim_{x \to a} \frac{f ( x )}{g ( x )} = \frac{ℓ}{m}$ , provided $m \neq = 0$ .
Inequality: if $f (x) \leq g (x)$ near $a$ , then $ℓ \leq m$ .

These rules extend to any finite number of functions. They are the reason we can evaluate limits of polynomials by simple substitution: sums and products of continuous functions behave predictably.

Caveat. These rules require both

ℓ

and

m

to be finite. If either limit is infinite or fails to exist, we cannot apply these rules blindly; we must first resolve the indeterminate form.

7. Limit of a Composite Function

If $f$ is continuous at $x = lim_{x \to a} g (x)$ , then

x \to a lim f (g (x)) = f (x \to a lim g (x)) .

In words: we can move the limit inside a continuous outer function. This is extremely useful for logarithms, exponentials, roots, and trigonometric outer functions.

Solved Example 4

lim_{x \to a} f (x) = ℓ > 0

, evaluate

lim_{x \to a} ln f (x)

Solution:

Since $ln$ is continuous at $ℓ > 0$ , we pull the limit inside:

x \to a lim ln f (x) = ln (x \to a lim f (x)) = ln ℓ .

8. Limits at Infinity: Polynomial Ratios

For rational functions with polynomial numerator and denominator, the behaviour as $x \to \infty$ is controlled entirely by the highest-degree terms.

x \to \infty lim \frac{a _{0} x ^{m} + a _{1} x ^{m - 1} + \dots}{b _{0} x ^{n} + b _{1} x ^{n - 1} + \dots} = ⎩ ⎨ ⎧ \frac{a _{0}}{b _{0}}, m = n 0, m < n \infty, m > n and \frac{a _{0}}{b _{0}} > 0 - \infty, m > n and \frac{a _{0}}{b _{0}} < 0

Also useful:

x \to \infty lim a^{x} = ⎩ ⎨ ⎧ 0, 0 \leq a < 1 1, a = 1 \infty, a > 1 does not exist, a < 0.

Solved Example 5

Let

f (x) = \frac{a x + b}{x + 1}

. Given

lim_{x \to 0} f (x) = 2

and

lim_{x \to \infty} f (x) = 1

, find

f (- 2)

Solution:

From the first condition (direct substitution at $x = 0$ ):

x \to 0 lim \frac{a x + b}{x + 1} = \frac{b}{1} = 2 ⟹ b = 2.

From the second condition (degrees equal, so ratio of leading coefficients):

x \to \infty lim \frac{a x + b}{x + 1} = \frac{a}{1} = 1 ⟹ a = 1.

Hence $f (x) = \frac{x + 2}{x + 1}$ , so $f (- 2) = \frac{0}{- 1} = 0$ .

Common Mistakes to Avoid

Watch out

Confusing $lim_{x \to a} f (x)$ with $f (a)$ . The limit is about the approach to $a$ ; the value $f (a)$ can be different, or $f (a)$ may not even be defined. These two quantities agree only when $f$ is continuous at $a$ .
Concluding a limit exists after checking only one side. For any point where the function's rule changes (piecewise definitions, absolute values, greatest integer), you must check LHL and RHL separately.
Treating indeterminate forms as if they equal a number. Writing " $\frac{0}{0} = 1$ " or " $\infty - \infty = 0$ " is wrong. These are labels for situations that need further work, not values.
Applying algebra of limits with infinite limits. The rules $lim (f + g) = lim f + lim g$ and its friends require both limits to be finite. Using them when one limit is $\infty$ can give nonsense.
Forgetting that $x \to a$ means $x \neq = a$ . When simplifying $\frac{( x - a ) ( x - b )}{x - a}$ , you can cancel $(x - a)$ because $x \neq = a$ throughout the limit process.
Assuming $1^{\infty} = 1$ . This is one of the seven indeterminate forms; the classic counter-example is $lim_{x \to \infty} (1 + 1/ x)^{x} = e$ , not $1$ .

Frequently Asked Questions

Q1. What is the difference between $lim_{x \to a} f (x)$ and $f (a)$ ?

The limit $lim_{x \to a} f (x)$ is the value $f (x)$ approaches as $x$ gets close to $a$ (but $x \neq = a$ ). The number $f (a)$ is the actual value of the function at $a$ . They may be different, and $f (a)$ may not exist at all while the limit still exists. When they agree (both exist and are equal), the function is called continuous at $a$ .

Q2. When does $lim_{x \to a} f (x)$ fail to exist?

The two-sided limit fails to exist in three main situations: (i) the LHL and RHL are both finite but unequal (jump); (ii) at least one of the one-sided limits is $\pm \infty$ (vertical asymptote); (iii) the function oscillates without settling near $a$ , as with $sin (1/ x)$ near $x = 0$ .

Q3. Why is $\frac{0}{0}$ called indeterminate instead of undefined?

Undefined means "no answer." Indeterminate means "not enough information yet." The limits $lim_{x \to 0} \frac{x}{x} = 1$ , $lim_{x \to 0} \frac{x ^{2}}{x} = 0$ , and $lim_{x \to 0} \frac{x}{x ^{3}}$ are all of the form $\frac{0}{0}$ but give completely different answers. The form alone does not decide the value; the specific functions do.

Q4. Is the limit at infinity a real number?

If it exists as a finite value, yes. For example, $lim_{x \to \infty} \frac{1}{x} = 0$ . When we write $lim_{x \to \infty} f (x) = \infty$ , we mean the limit does not exist as a finite real number; the function grows without bound. In JEE contexts we usually distinguish finite limits from $\pm \infty$ carefully.

Q5. Do I need the $ε$ - $δ$ definition for JEE problems?

JEE problems are almost always solved using the working intuition plus algebra, standard limits, expansions, or L'Hospital's rule. The $ε$ - $δ$ definition is rarely tested directly, but understanding it deepens your grip on why limit rules work, which helps in trickier problems involving one-sided behaviour and existence.

Q6. How do I decide whether to compute LHL and RHL separately?

Compute them separately whenever the function's expression changes at $x = a$ . Common triggers: piecewise functions, absolute values around the point (like $∣ x - a ∣$ ), the greatest integer function $[x]$ , the fractional part ${x}$ , and step or sign functions. For smooth expressions (polynomials, sines, exponentials), one calculation suffices.

Q7. Can a limit exist if $f$ is not defined at the point?

Yes. This is one of the most powerful features of limits. For example, $f (x) = \frac{x ^{2} - 1}{x - 1}$ is undefined at $x = 1$ (denominator is zero), yet $lim_{x \to 1} f (x) = 2$ because $f (x) = x + 1$ for all $x \neq = 1$ .

Q8. What is the relationship between limits and continuity?

A function $f$ is continuous at $x = a$ if and only if all three quantities exist and agree: LHL, RHL, and $f (a)$ . So continuity is limit existence plus the additional condition that the limit equals the function's value. This is the bridge from this concept to the continuity chapter.

Previous year questions on Introduction To Limits

1 question from past papers, each with a step-by-step solution.

JEE Main 2025 Jan 24 Shift 1, Mathematics Q3

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1. What is a Limit? The Intuitive Idea

2. Formal Definition of a Limit

2.1 Informal (working) definition

2.2 Formal ε-δ definition

3. One-Sided Limits: LHL and RHL

3.1 Right-Hand Limit (RHL)

3.2 Left-Hand Limit (LHL)

4. Existence of a Limit

5. Indeterminate Forms

6. Algebra of Limits

7. Limit of a Composite Function

8. Limits at Infinity: Polynomial Ratios

Common Mistakes to Avoid

Frequently Asked Questions

Q1. What is the difference between limx→a​f(x) and f(a)?

Q2. When does limx→a​f(x) fail to exist?

Q3. Why is 00​ called indeterminate instead of undefined?

Q4. Is the limit at infinity a real number?

Q5. Do I need the ε-δ definition for JEE problems?

Q6. How do I decide whether to compute LHL and RHL separately?

Q7. Can a limit exist if f is not defined at the point?

Q8. What is the relationship between limits and continuity?

Previous year questions on Introduction To Limits

Ready to master Limits, Continuity And Differentiability?

2.2 Formal $ε$ - $δ$ definition

Q1. What is the difference between $lim_{x \to a} f (x)$ and $f (a)$ ?

Q2. When does $lim_{x \to a} f (x)$ fail to exist?

Q3. Why is $\frac{0}{0}$ called indeterminate instead of undefined?

Q5. Do I need the $ε$ - $δ$ definition for JEE problems?

Q7. Can a limit exist if $f$ is not defined at the point?