Limits

Definition of a Limit

In calculus, the definition of a limit helps us understand how a function behaves as its input approaches a specific point. It tells us the value that $f(x)$ gets closer and closer to when $x$ moves towards a particular number, say $a$. The limit of a function is denoted as $\lim_{x \to a} f(x)$ and is a foundational idea behind continuity and differentiation.
When the value of $f(x)$ approaches a single, finite number as $x$ approaches $a$, we say that the limit exists. However, if $f(x)$ jumps, oscillates, or tends to infinity near that point, then the limit does not exist.

Understanding Limit Notation and Terminology

In most problems, you’ll see expressions like $\lim_{x \to 3} f(x)$ or $\lim_{x \to \infty} f(x)$.
Here’s what each part means:

• $x \to 3$ indicates the variable $x$ is approaching 3 (not necessarily equal to it).

• $f(x)$ is the function whose behavior we’re studying near that point.

• The overall expression tells us what value $f(x)$ is approaching as $x$ nears 3.

Other common notations include:

• Left-hand limit: $\lim_{x \to a^-} f(x)$ – $x$ approaches $a$ from the left side.

• Right-hand limit: $\lim_{x \to a^+} f(x)$ – $x$ approaches $a$ from the right side.

For the overall limit to exist, both Left-Hand and Right-Hand Limits must be equal.

When Does a Limit Not Exist?

A limit does not exist in the following cases:

1. Different Left and Right Limits:
   If $\lim_{x \to a^-} f(x) \ne \lim_{x \to a^+} f(x)$, the function jumps near $x = a$.

2. Infinite Behavior:
   If $f(x)$ grows without bound as $x$ approaches $a$, such as $\lim_{x \to 0} \frac{1}{x}$, the limit is said to diverge to infinity.

3. Oscillating Functions:
   When $f(x)$ keeps fluctuating between values near a point (for example, $\lim_{x \to 0} \sin \frac{1}{x}$), no single value is approached.

Understanding when limits fail to exist is just as important as finding them — it helps you analyze discontinuities, asymptotic behavior, and non-differentiable points in calculus.

Similarly, when $x$ approaches $2$, the value of $f(x)$ approaches $4$, i.e. $\lim\limits_{x \to 2} f(x) = 4$ or $\lim\limits_{x \to 2} x^2 = 4$.

In general, as $x \to a$ , $f(x) \to l$, then $l$ is called the limit of the function $\mathrm{f}(\mathrm{x})$, which is symbolically written as $\lim\limits_{x \to a} f(x) = l$.

Now consider the function

$f(x)=\frac{x^2-6 x-7}{x-7}$
We can factor the function as shown
$f(x)=\frac{(x-7)(x+1)}{x-7} \quad$ [Cancel like factors in numerator and denominator.]

$f(x)=x+1, x \neq 7$

Notice that $x$ cannot be $7$, or we would be dividing by $ 0$ , so $7$ is not in the domain of the original function. To avoid changing the function when we simplify, we set the same condition, $x \neq 7$, for the simplified function. We can represent the function graphically

What happens at $x=7$ completely differs from what happens at points close to $x=7$ on either side. Just observe that as the input $x$ approaches $7$ from either the left or the right, the output approaches $8$. The output can get as close to $8$ as we like if the input is sufficiently near $7$ . So we say that the limit of this function at $x=7$ equals $8$.

So even if the function does not exist at x = a, still the limit can exist at that point as the limit is concerned only about the points close to $x=a$ and NOT at $x=a$ itself.

Types of Limits in Calculus

In calculus, not all limits behave the same way. Depending on how $x$ approaches a point and how the function behaves near that point, we can categorize limits into different types. Understanding these types of limits — one-sided limits, infinite limits, and limits at infinity — helps build a strong foundation for topics like continuity, asymptotes, and convergence.
Let’s explore each type in detail.

One-Sided Limits (Left-Hand and Right-Hand Limits)

A one-sided limit studies how a function behaves as the input approaches a number from one side only — either from the left or from the right.
This concept is crucial when dealing with functions that have jumps or sharp corners.

• The left-hand limit (LHL) of $f(x)$ as $x$ approaches $a$ is written as:
  $\lim_{x \to a^-} f(x)$
  It means we are observing how $f(x)$ behaves as $x$ approaches $a$ from values less than $a$.

• The right-hand limit (RHL) of $f(x)$ as $x$ approaches $a$ is written as:
  $\lim_{x \to a^+} f(x)$
  It describes how $f(x)$ behaves as $x$ approaches $a$ from values greater than $a$.

For the limit $\lim_{x \to a} f(x)$ to exist, both one-sided limits must exist and be equal, i.e.,
$\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)$.

If they are different, we say the limit does not exist at that point.
This situation often occurs at jump discontinuities in piecewise functions.

Infinite Limits and Limits at Infinity

Sometimes, as $x$ approaches a certain value, the function grows without bound — either positively or negatively. These are called infinite limits.
For example:
$\lim_{x \to 0^+} \frac{1}{x} = +\infty$ and $\lim_{x \to 0^-} \frac{1}{x} = -\infty$.

An infinite limit means the function’s values increase or decrease without limit near a particular $x$ value. While the limit doesn’t exist in the finite sense, it tells us about the vertical asymptote of the graph.

On the other hand, a limit at infinity looks at what happens to $f(x)$ as $x$ itself becomes infinitely large (or small).
For example:
$\lim_{x \to \infty} \frac{1}{x} = 0$.
Here, as $x$ increases without bound, $f(x)$ approaches zero.
These limits help describe horizontal asymptotes of a function and are widely used in studying rational and exponential functions.

Limits of Sequences vs. Limits of Functions

While limits of functions deal with continuous inputs ($x$ values approaching any real number), limits of sequences apply to discrete situations where the variable takes integer values.

A sequence limit examines what happens to $a_n$ as $n$ becomes very large.
It’s written as:
$\lim_{n \to \infty} a_n = L$
This means the terms of the sequence get closer and closer to $L$ as $n$ increases.

For example:
If $a_n = \frac{1}{n}$, then $\lim_{n \to \infty} a_n = 0$.
But if $a_n = (-1)^n$, the sequence oscillates and has no limit.

While limits of functions describe the continuous behavior of curves, limits of sequences describe the convergence or divergence of number lists.

How to Find Limits

Finding the limit of a function is one of the most common and important tasks in calculus. There are several techniques you can use, depending on the form of the function. From direct substitution to L’Hôpital’s Rule, each method helps simplify the expression so you can determine the value the function approaches. Let’s look at the most effective ways to evaluate limits.

Evaluating Limits by Substitution and Simplification

The easiest method to find a limit is direct substitution.
If the function is continuous at the point you’re evaluating, simply plug in the value of $x$.

For example: $\lim_{x \to 2} (3x + 1) = 3(2) + 1 = 7$.

However, sometimes direct substitution leads to indeterminate forms like $\frac{0}{0}$ or $\frac{\infty}{\infty}$.
In such cases, you can simplify the expression by canceling common factors, expanding, or using algebraic manipulation.

Example: $\lim_{x \to 1} \frac{x^2 - 1}{x - 1}$
Factorizing the numerator gives $(x - 1)(x + 1)$, so the limit becomes:
$\lim_{x \to 1} (x + 1) = 2$.

Substitution and simplification are the starting points for most limit problems — always try these before moving to advanced techniques.

Factorization and Rationalization Techniques

When substitution gives $\frac{0}{0}$, factorization or rationalization often resolves the indeterminate form.

1. Factorization Method:
   Example: $\lim_{x \to 3} \frac{x^2 - 9}{x - 3}$
   Factor the numerator: $(x - 3)(x + 3)$
   Cancelling $(x - 3)$, we get $\lim_{x \to 3} (x + 3) = 6$.

2. Rationalization Method:
   This is useful when square roots are involved.
   Example: $\lim_{x \to 0} \frac{\sqrt{x + 1} - 1}{x}$
   Multiply numerator and denominator by the conjugate $\sqrt{x + 1} + 1$:
   $\frac{(\sqrt{x + 1} - 1)(\sqrt{x + 1} + 1)}{x(\sqrt{x + 1} + 1)} = \frac{x}{x(\sqrt{x + 1} + 1)} = \frac{1}{\sqrt{x + 1} + 1}$
   So, $\lim_{x \to 0} \frac{\sqrt{x + 1} - 1}{x} = \frac{1}{2}$.

These methods simplify expressions so you can easily apply substitution afterward.

Using L’Hôpital’s Rule for Indeterminate Forms

When algebraic techniques fail, L’Hôpital’s Rule comes to the rescue.
It applies to limits that result in indeterminate forms such as $\frac{0}{0}$ or $\frac{\infty}{\infty}$.

If $\lim_{x \to a} \frac{f(x)}{g(x)}$ gives $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then:
$\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$, provided the latter limit exists.

Example: $\lim_{x \to 0} \frac{\sin x}{x} = \lim_{x \to 0} \frac{\cos x}{1} = 1$.

L’Hôpital’s Rule is one of the most powerful techniques for solving complex limits, especially in calculus exams like JEE, CUET, or IIT JAM.

Tricks to Solve Limits Faster for Competitive Exams

To solve limits quickly under exam pressure, remember these smart shortcuts:

• Use standard limits like
  $\lim_{x \to 0} \frac{\sin x}{x} = 1$ and $\lim_{x \to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2}$.

• Recognize patterns that match exponential or logarithmic limits, e.g.
  $\lim_{x \to 0} (1 + x)^{1/x} = e$.

• Always simplify step-by-step before applying formulas.

• Check whether the function is continuous at the point; if yes, direct substitution saves time.

• For rational functions, divide numerator and denominator by the highest power of $x$ to find limits at infinity.

Limit Laws and Theorems

Limit laws provide the essential rules for simplifying and combining limits systematically. These laws are the foundation for proving continuity and differentiability.

Basic Limit Laws – Sum, Product, and Quotient

If $\lim_{x \to a} f(x) = L_1$ and $\lim_{x \to a} g(x) = L_2$, then:

1. Sum Law: $\lim_{x \to a} [f(x) + g(x)] = L_1 + L_2$

2. Difference Law: $\lim_{x \to a} [f(x) - g(x)] = L_1 - L_2$

3. Product Law: $\lim_{x \to a} [f(x) \cdot g(x)] = L_1 \cdot L_2$

4. Quotient Law: $\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{L_1}{L_2}$, where $L_2 \ne 0$

5. Power Law: $\lim_{x \to a} [f(x)]^n = (L_1)^n$

These limit laws make it easier to break complicated expressions into smaller, manageable parts.

The Squeeze (Sandwich) Theorem Explained

The Squeeze Theorem, also known as the Sandwich Theorem, helps when a function is “trapped” between two other functions whose limits are known.

If $f(x) \leq g(x) \leq h(x)$ for all $x$ near $a$, and
$\lim_{x \to a} f(x) = \lim_{x \to a} h(x) = L$,
then $\lim_{x \to a} g(x) = L$.

Example:
Since $-1 \le \frac{\sin x}{x} \le 1$ for $x$ near $0$, we use the squeeze theorem to show
$\lim_{x \to 0} \frac{\sin x}{x} = 1$.

This theorem is especially useful for trigonometric functions and oscillating functions.

Connection Between Limit and Continuity

A function is continuous at $x = a$ if three conditions hold:

1. $f(a)$ is defined.

2. $\lim_{x \to a} f(x)$ exists.

3. $\lim_{x \to a} f(x) = f(a)$.

This means that the value of the function and the limit at that point are the same, there’s no sudden jump or hole in the graph.
So, continuity is directly built on the concept of limits. Without understanding limits, it’s impossible to grasp continuity, derivatives, or the smoothness of curves.

Some Properties of Limits:

• $\lim\limits_{\mathrm{x} \to \mathrm{a}} \mathrm{c} = \mathrm{c}$, where $c$ is a constant quantity.
• The value of $\lim\limits_{\mathrm{x} \to \mathrm{a}} {x} = \mathrm{a}$.
• Value of $\lim\limits_{x \to a} (b x + c) = b a + c$.
• $\lim\limits_{x \to a} x^n = a^n$, if $n$ is a positive integer.

• Limit of a Constant:
  $\lim_{x \to a} c = c$
  The limit of a constant is always the constant itself.

• Identity Law:
  $\lim_{x \to a} x = a$
  The limit of the variable as it approaches $a$ is $a$.

• Sum Law:
  $\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)$
  The limit of a sum equals the sum of the limits.

• Difference Law:
  $\lim_{x \to a} [f(x) - g(x)] = \lim_{x \to a} f(x) - \lim_{x \to a} g(x)$
  The limit of a difference equals the difference of the limits.

• Product Law:
  $\lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)$
  The limit of a product equals the product of the limits.

• Quotient Law:
  $\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$, where $\lim_{x \to a} g(x) \ne 0$
  The limit of a quotient equals the quotient of the limits (if the denominator limit is non-zero).

• Power Law:
  $\lim_{x \to a} [f(x)]^n = [\lim_{x \to a} f(x)]^n$
  Valid for any real number $n$.

• Root Law:
  $\lim_{x \to a} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x \to a} f(x)}$, if $\lim_{x \to a} f(x) \ge 0$ for even $n$
  Applies to nth roots under proper domain restrictions.

• Constant Multiple Law:
  $\lim_{x \to a} [k \cdot f(x)] = k \cdot \lim_{x \to a} f(x)$
  A constant can be factored out of the limit.

• Squeeze (Sandwich) Theorem:
  If $f(x) \le g(x) \le h(x)$ for all $x$ near $a$, and
  $\lim_{x \to a} f(x) = \lim_{x \to a} h(x) = L$,
  then $\lim_{x \to a} g(x) = L$.

• Limits of Composite Functions:
  If $\lim_{x \to a} g(x) = L$ and $f$ is continuous at $L$, then
  $\lim_{x \to a} f(g(x)) = f(\lim_{x \to a} g(x)) = f(L)$.

• Limits of Absolute Value:
  $\lim_{x \to a} |f(x)| = |\lim_{x \to a} f(x)|$

Some Useful Limits

This section highlights commonly used limits in calculus that appear frequently in exams and problem-solving.

Solved Examples Based on Limits

Example 1: Which of the following is incorrect?

(1) As $x$ approaches $2$, tends to reach $4$.
2) As $x$ approaches tends to reach $0$.
3) As $x$ approach tends to reach $\infty$.

4) As $x$ approach $\frac{\pi}{2}$, then $\tan x$ has a tendency to reach $\infty$.

Solution:
Limits describe the behaviour of a function $f(x)$ as its variable $x$ approaches a particular number.

1) $\lim\limits_{x \to 2} x^2 = 4$ — Statement 1 is correct.

2) $\lim\limits_{x \to \pi} \sin x = 0$ — Statement 2 is correct.

3) $\lim\limits_{x \to \pi / 2} \sin x = 1$ — Statement 3 is incorrect.

4) $\lim\limits_{x \to \pi / 2} \tan x = \infty$ — Statement 4 is correct.
Hence, the answer is the option 3. $\qquad$

Example 2: $\lim _{x \rightarrow 0}\left(\left(\frac{\left(1-\cos ^2(3 x)\right.}{\cos ^3(4 x)}\right)\left(\frac{\sin ^3(4 x)}{\left(\log _e(2 x+1)^5\right.}\right)\right)$ is equal to [JEE MAINS 2023]
1) $24$
2) $9$
3) $18$
4) $15$

Solution:

$
\begin{aligned}
& \lim _{x \rightarrow 0}\left[\frac{1-\cos ^2 3 x}{9 x^2}\right] \frac{9 x^2}{\cos ^3 4 x} \cdot \frac{\left(\frac{\sin 4 x}{4 x}\right)^3 \times 64 x^3}{\left[\frac{\ln (1+2 x)}{2 x}\right]^5 \times 32 x^5} \\
& \lim _{x \rightarrow 0} 2\left(\frac{1}{2} \times \frac{9}{1} \times \frac{1 \times 64}{1 \times 32}\right)=18
\end{aligned}
$
Hence, the answer is the option (3).

Example 3: Let $a_1, a_2, a_3, \ldots, a_n n$ be $n$ positive consecutive terms of an arithmetic progression. If this is its common difference, then: $\lim\limits _{n \rightarrow \infty} \sqrt{\frac{d}{n}}\left(\frac{1}{\sqrt{a_1}+\sqrt{a_2}}+\frac{1}{\sqrt{a_2}+\sqrt{a_3}}+\ldots \ldots+\frac{1}{\sqrt{a_{n-1}}+\sqrt{a_n}}\right)$ is

[JEE MAINS 2023]
1) $\frac{1}{\sqrt{d}}$
2) $1$
3) $\sqrt{d}$
4) $0$

Solution:

$
\begin{aligned}
& \lim\limits_{n \to \infty} \sqrt{\frac{d}{n}} \left( \frac{\sqrt{a_1} - \sqrt{a_2}}{a_1 - a_2} + \frac{\sqrt{a_2} - \sqrt{a_3}}{a_2 - a_3} + \cdots + \frac{\sqrt{a_{n-1}} - \sqrt{a_n}}{a_{n-1} - a_n} \right) \\
& = \lim\limits_{n \to \infty} \sqrt{\frac{d}{n}} \left( \frac{\sqrt{a_1} - \sqrt{a_2} + \sqrt{a_2} - \sqrt{a_3} + \cdots + \sqrt{a_{n-1}} - \sqrt{a_n}}{-d} \right) \\
& = \lim\limits_{n \to \infty} \sqrt{\frac{d}{n}} \left( \frac{\sqrt{a_n} - \sqrt{a_1}}{d} \right) \\
& = \lim\limits_{n \to \infty} \frac{1}{\sqrt{n}} \left( \frac{\sqrt{a_1 + (n - 1) d} - \sqrt{a_1}}{\sqrt{d}} \right) \\
& = \lim\limits_{n \to \infty} \frac{1}{\sqrt{d}} \left( \sqrt{\frac{a_1 + (n - 1) d}{n}} - \frac{\sqrt{a_1}}{n} \right) \\
& = 1
\end{aligned}
$

Hence, the answer is the option 2

Example 4: If $x$ approaches $2$, then the approximate value of is
1) $4$
2) $2$
3) $3$
4) $1$

Solution:

As we have learned

Condition on Limits -
The limit does not give actual value. It gives an approximate value.
- wherein
$f(x)=\frac{x^2+x-2}{x-1}$
$x$ is not defined at $\mathrm{x}=1$ but for $\mathrm{x}>1 \& \mathrm{x}<1$ it gives approximate values.

When x approaches $2, x-2$ simplifies to $\mathrm{x}+2$
Limit approaches to $4$
Hence, the answer is the option 1.

Example 5: If $x$ approaches $3$ , then $\frac{x^2-5 x+6}{x^2-4 x+3}$ has approximate value
1) $\frac{1}{2}$
2) $0$
3) $1$
4) $\frac{3}{2}$

Solution:

Condition on Limits -

The limit does not give actual value. It gives an approximate value.
- wherein

$
f(x)=\frac{x^2+x-2}{x-1}
$

$x$ is not defined at $x=1$ but for $x>1 \& x<1$ it gives approximate values.

$
\frac{x^2-5 x+6}{x^2-4 x+3}=\frac{(x-2)(x-3)}{(x-1)(x-3)}
$
When x approaches $3, \frac{x^2-5 x+6}{x^2-4 x+3}$ simplifies to $\frac{x-2}{x-1}$
$\therefore \frac{x-2}{x-1}$ approaches $\frac{3-2}{3-1}=1 / 2$
Hence, the answer is the option 1.

List of topics related to Limits

We have provided below the topics related to limits, to help you strengthen your concepts clearly.

Indeterminate Forms

Limits of Trigonometric Functions

Limits using expansion

Algebra of Limits

Higher Order Derivatives: Formula, Definition, Examples

NCERT Resources

Access complete NCERT notes, solutions, and exemplar problems for Limits and Derivatives in one place. These resources follow the latest CBSE guidelines and help build strong conceptual clarity for board exams and entrance tests.

NCERT Notes for Class 11 Maths Chapter 13 - Limits and Derivatives

NCERT Solutions for Class 11 Maths Chapter 13 - Limits and Derivatives

NCERT Exempar Solutions for Class 11 Maths Chapter 13 - Limits and Derivatives

Practice Questions based on Limits

Test your understanding with carefully selected limit questions for JEE, CUET, and board exams. These practice problems range from basic substitution to tricky indeterminate forms, helping you apply every concept you’ve learned effectively.

Limit - Practice Question MCQ

We have shared the links below to practice questions on the related topics of limits: