# Limits

** Limits** in mathematics are defined as a value approaching the output for the given input values of a function. Limits are used in calculus for finding the derivatives of the function. They are also used to define the continuity of the function.

The limit of any function is also used to find the integral of the function. The integral are of two types, Indefinite Integral, and Definite Integral, in definite integral we use the concept of upper limit and lower limit to find the answer to the definite integral. A function can reach a particular value from more than one path and the value of the function at that particular point is called the limit of the function at that point. Suppose we are given a function f(x) and when x approaches a the function approaches A this i represented using limit as:

lim_{ x}_{⇢ a }f(x) = A

In this article, we will learn the introduction to limits, properties of limits, limit and continuity, and others in detail.

Table of Content

## What Are Limits?

The limit of a function is defined as the unique real number that the functions take when the variable of the function approaches a particular point. For any given function f(x), and a real number ‘c’, the limit of the function is defined as,

lim_{x ⇢ a}f(x) = L

This is read as, “limit of f(x), as x approaches a equals L”

The image added below shows the function approaching the limit.

We can approach the limit of any function from two paths and the limit of the function exist only when both the individual limit of the function are equal. The two limits of the function are called Left Hand Limit(LHL) and the Right Hand Limit(RHL) of the function.

**Limits Definition**

**Limits Definition**

To define the limit of a function let us consider a real-valued function ** “f”** and the real number

**such that the variable of the function approaches the value “a” then the limit is normally defined as**

**“a”**

lim_{ x ⇢ a}f(x) = L

It is read as “the limit of f of x, as x approaches a equals L”.

### Formula Definition of Limit

The formula definition of the limit is given as,

For any function f(x) defined for all x ≠ aover an open interval containing a. Now suppose we have a real number L such that,

lim _{x ⇢ a} f(x) = L

Then for every ε > 0, there exists a δ > 0, such that, 0 < |x – a| < δ the,

**|f(x) – L| < ε**

**Types of Limits**

**Types of Limits**

Limits in maths are of several types, each describing different situations and behaviors of functions as the independent variable approaches a certain value or infinity. Here are the main types of limits:

### One-Sided Limits

There are two path to approach any point in 2D space along a curve. That are from Left Hand Side of Curve or Right Hand Side of Curve. Approaching the curve from either sides allow us to find two separate limit of the function. These two limits are called,

- Left Hand Limit
- Right Hand Limit

Now, let’s learn about them in detail.

**LHL (Left Hand Limit)**

**LHL (Left Hand Limit)**

The the left-hand limit of the function is defined as the vale of the function when the variable approaches the value from left side of the function. It is represented as,

lim_{x ⇢ a-}f(x) = L

**RHL (Right Hand Limit)**

**RHL (Right Hand Limit)**

The the right-hand limit of the function is defined as the vale of the function when the variable approaches the value from right side of the function. It is represented as,

lim_{x ⇢ a+ }f(x) = L

### Two-Sided Limits

Two-sided limits, also known as bilateral limits, are a fundamental concept in calculus that describe the behavior of a function as the independent variable approaches a particular value from both the left and the right sides simultaneously.

Formally, let ** f(x)** be a function defined on an open interval containing

*=*

*x**, except possibly at*

*c**=*

*x**itself. The two-sided limit of*

*c***as**

**f(x)***approaches*

*x**, denoted as:*

*c***lim**_{x→c}** f(x)**

exists if and only if both the left-hand limit (as * x* approaches

*from the left) and the right-hand limit (as*

*c**approaches*

*x**from the right) exist and are equal.*

*c*Mathematically, the two-sided limit * L* of

**as**

**f(x)***approaches*

*x**is defined as follows:*

*c***lim**_{x→c}** f(x) = L**

if for every positive number * ε*, there exists a positive number

*such that if*

*δ*

**0 < ∣**

**x**

**−**

**c**

**∣ <***, then*

**δ**

**∣**

**f**

**(**

**x**

**) −**

**L**

**∣ <***.*

**ε**In simpler terms, this definition means that as * x* gets arbitrarily close to

*,*

*c**(*

*f**) gets arbitrarily close to*

*x**.*

*L*The concept of two-sided limits is essential for understanding continuity, determining the existence of limits at a point, and evaluating derivatives in calculus. It helps analyze the behavior of functions near specific points and understand their overall characteristics and behavior in various mathematical contexts.

**Infinite Limits**

**Infinite Limits**

Infinite limits occur when the value of a function approaches positive or negative infinity as the independent variable approaches a particular point. Formally, if the value of f(x) becomes arbitrarily large (positive or negative) as * x* approaches a certain value

*, the limit is said to be infinite.*

*c*: If f(x) increases without bound as**Positive Infinite Limit**approaches*x*, the limit is denoted as*c***lim**_{x→c }.**f(x) = +∞**: If f(x) decreases without bound as**Negative Infinite Limit**approaches*x*, the limit is denoted as*c***lim**_{x→c}.**f(x) = −∞**

For instance, consider the function **f(x) = 1/x**^{2}** **. As

*approaches 0 from either the positive or negative direction,*

*x***becomes increasingly large (approaches infinity), so the limit of**

**f(x)****as**

**f(x)***approaches 0 is +∞.*

*x***Limits at Infinity**

**Limits at Infinity**

Limits at infinity describe the behavior of a function as the independent variable grows without bound (approaches positive or negative infinity).

: If f(x) approaches a finite limit as**Limit at Positive Infinity**goes to positive infinity, it is denoted as*x***lim**_{x→+∞}.**f(x) = L**: If f(x) approaches a finite limit as**Limit at Negative Infinity**goes to negative infinity, it is denoted as*x***lim**_{x→−∞}.**f(x) = L**

For example, consider the function ** f(x) = 1/x**. As

*grows without bound (either positively or negatively),*

*x***approaches 0. Thus,**

**f(x)**

**lim**

_{x}

_{→+∞}

**1/**

**x****and**

** = 0**

**lim**

_{x}

_{→−∞}**.**

**1/x = 0**Understanding infinite limits and limits at infinity is crucial in calculus, particularly when analyzing the long-term behavior of functions, determining horizontal asymptotes, and evaluating integrals involving unbounded intervals. These concepts help in understanding the behavior of functions as the independent variable approaches extreme values.

**Properties of Limits**

**Properties of Limits**

Various properties of the limit of a function are,

- lim
_{ x ⇢ a}k = k, where k is a constant quantity - The value of lim
_{ x ⇢ a}x = a - Value of lim
_{ x ⇢ a}bx + c = ba + c - lim
_{x ⇢ a}x^{n}= a^{n}if n is a positive integer. - Value of lim
_{x ⇢ +0}1/x^{r}= +∞ - lim
_{x ⇢ −0}1/x^{r}= −∞, if r is odd - lim
_{x ⇢ −0}1/x^{r}= +∞, if r is even

## Algebra of Limit

Algebra of the limit of the function are added below,

** Law of Addition:** lim

_{x⇢a}{f(x) + g(x)} = lim

_{x⇢a}f(x) + lim

_{x⇢a}g(x)

** Law of Subtraction:** lim

_{x⇢a }{f(x) – g(x)} = lim

_{x⇢a}f(x) – lim

_{x⇢a}g(x)

** Law of Multiplication:** lim

_{x⇢a}{f(x) . g(x)} = lim

_{x⇢a}f(x) . lim

_{x⇢a}g(x)

** Law of Division:** lim

_{x⇢a}{f(x) / g(x)} = lim

_{x⇢a}f(x) / lim

_{x⇢a}g(x)

### Special Rules of Limit

Various rules that are used to simplify the limit of the function are,

**lim**_{x⇢a}** (x**^{n}** – a**^{n}**)/(x – a) = na**^{(n-1)}

**lim**_{x⇢a}** sin x/x = 1**

**lim**_{x⇢a}** tan x/x = 1**

**lim**_{x⇢a }**(1 – cos x)/x = 0**

**lim**_{x⇢a}** cos x = 1**

**lim**_{x⇢a}** e**^{x}** = 1**

**lim**_{x⇢a}** (e**^{x}** – 1)/x = 1**

**lim**_{x⇢a}** (1 + 1/x)**^{x}** = e**

## Limits and Functions

Limit of any function is defined as the value of the function when the independent variable of the function approaches a particular value. A function’s limit exist only when the left hand limit and right hand limit of the function both exist and are equal.

Left hand Limit (LHL) of the function is the limit of the function when the limit of the function approaches from the left side of the function.

Right hand Limit (RHL) of the function is the limit of the function when the limit of the function approaches from the right side of the function.

**Limit of Polynomial Function**

**Limit of Polynomial Function**

Limit of the polynomial function are added below, consider a polynomial function,

f(x) = a_{0} + a_{1}x + a_{2}x^{2} + … + a_{n}x^{n}

Here, a_{0}, a_{1}, … , a_{n} are all constants. At any point x = a, the limit of this polynomial function is

lim _{x ⇢ a }f(x) = lim_{ x ⇢ a }[a_{0} + a_{1}x + a_{2}x^{2 }+ . . . + a_{n}x^{n}]

= lim _{x ⇢ a }a_{0} + a_{1}lim _{x ⇢ a} x + a_{2}lim _{x ⇢ a} x^{2} + . . . + a_{n}lim _{x ⇢ a} x^{n}

lim_{ x ⇢ a } a_{0} + a_{1}a + a_{2}a^{2} + . . . + a_{n}a^{n}

lim _{x ⇢ a} = f(a)

**Limit of Rational Function**

**Limit of Rational Function**

The limit of any rational function of the type m(x)/n(x), where n(x) ≠ 0 and m(x) and n(x) are polynomial functions, is:

= lim _{x ⇢ a} [m(x)/n(x)]

= lim _{x ⇢ a} m(x)/lim_{ x ⇢ a }n(x)

= m(a)/m(b)

The very first step to find the limit of a rational function is to check if it is reduced to the form 0/0 at some point. If it is so, then we need to do some adjustments so that one can calculate the value of the limit. This can be done by canceling the factor which causes the limit to be of the form 0/0. For example,

f(x) = (x^{2}– 4x + 4)/(x^{2}− 4)Taking limit over it for x = 2, the function is of the form 0/0,

lim

_{x ⇢ 2}f(x)= lim

_{x ⇢ 2}(x^{2}– 4x + 4)/(x^{2}− 4)= lim

_{ x ⇢ 2}[( x – 2)^{2}/(x + 2)( x – 2)]= lim

_{ x ⇢ 2}[(x – 2)/(x + 2)]= 0/4 ( ≠ 0/0 )

= 0

## Limits of Complex Functions

If we are given a complex function then the limit of the complex function is calculated as, suppose we are given a function f(z) where z is a complex variable then the z = z_{0 }then the f(z) is differentiable if,

lim_{Δz→0}[f(z_{0}+ Δz) – f(z_{0})]/Δzwhere, Δz = Δx + iΔy

## Limits of Exponential Functions

The limit of exponential function is easily calculated by taking into consideration the initial value of the exponential function. Suppose we are given an exponential function f(x) = a^{x} where a > 0.

For f(b) > 1

- lim
_{x→∞ }a^{x}= ∞ - lim
_{x→-∞}a^{x}= 0

For 0 < f(b) < 1

- lim
_{x→∞}a^{x}= 0 - lim
_{x→-∞}a^{x}= ∞

## Limit of a Function of Two Variables

For the given function with two variables say f(x, y) then suppose if the limit of the function is C, (x, y) → (a, b) provided that ϵ > 0 here exists Δ > 0 such that |f(x, y) – C| < ϵ whenever 0 < √{(x -a)^{2} + (y – b)^{2}} < Δ. Then,

Iim_{(x, y) → (a, b)}f(x, y) = C

## Calculating Limits

The limit of the function is calculated by following the techniques added below,

### Direct Substitution

In direct substitution we directly put the value of the limit in the function as the limit of the function is defined at that point. For example,

lim _{x → 4} 3x + 2

= 3(4) + 2 = 12 + 2 = 14

### Factoring and Simplifying

Sometimes we have to factorize to find the limit of the function as limit of the function does not exist in case of direct substitution. For example,

lim _{x → 2} (x^{2} – 4)/(x – 2)

here directly substituting x = 2 results in indeterminant form (0/0). So we first simplify the limit as,

lim _{x → 2} (x – 2)(x + 2)/(x – 2)

lim _{x → 2} (x + 2)

Now, taking the limit

2 + 2 = 4

### Rationalizing Techniques

Sometimes we have to rationalize the denominator or numerator of the limit with its conjugate to simplify the limit. For example,

lim _{x → 3} (x – 3)/(x^{2} – 9)

lim _{x → 3} (x – 3)/(x^{2} – 9) × (x + 3)/(x + 3)

lim _{x → 3} (x^{2} – 9)/(x^{2} – 9)(x + 3)

lim _{x → 3} 1/(x + 3)

1/(3 + 3) = 1/6

### L’Hospital Rule

If all the above techniques fails to find the limit of the function or finding the limit is not easy with these techniques then we use the L’Hôpital Rule. This rule states that, the limit of any function (given as f(x)/g(x)) is equal to the limit of function where we take the derivative of both numerator and denominator. This rule only works when the limit of the function is in ** 0/0** or

**form. We can differentiate both numerator and denominator simultaneously till the indeterminant form disappears. Now L’Hôpital Rule is represented as,**

**∞/∞**

lim_{x → a}f(x)/g(x) = lim_{x → a}f'(x)/g'(x) = lim_{x → a}f”(x)/g”(x)

For example,

lim_{x → ∞}(x + 3)/(x + 2)Here directly taking the limit we get, ∞/∞ form. (Using L’Hopital Rule)

lim

_{x → ∞}1/1 = 1

| |
---|---|

**Examples of Limits**

**Examples of Limits**

**Example 1: lim **_{x ⇢ 6}** x/3 **

**Solution:**

lim

_{ x ⇢ 6}x/3= 6/3

= 2

**Example 2: lim **_{x ⇢ 2}** (x**^{2}** – 4)/(x – 2)**

**Solution:**

(x

^{2}– 4) = (x^{2}– 2^{2}) = ( x – 2 )( x – 2 )= lim

_{ x ⇢ 2}(x^{2}– 4)/(x – 2)= lim

_{x ⇢ 2}(x- 2)(x + 2)/(x – 2)= lim

_{x ⇢ 2}(x + 2)= 4

**Example 3: lim **_{x ⇢ 1/2 }**(2x – 1)/(4x**^{2}** – 1)**

**Solution:**

4x

^{2}– 1 = (2x^{2}) – (1^{2}) = (2x + 1) (2x – 1)= lim

_{x ⇢ 1/2 }(2x – 1)/(4x^{2}– 1)= lim

_{ x ⇢ 1/2 }(2x- 1)/(2x – 1) (2x + 1)= lim

_{x ⇢ 1/ 2 }1/(2x + 1)= 1/{2 × (1/2) + 1}

= 1/2

## Practice Questions on Limits

**Q1. Find the value of lim**_{ x →2}** 2x**^{2 }**+ 4x + 11**

**Q2. Find the value of lim **_{x →-1}** x**^{2}** + x – 4**

**Q3. Find the value of lim **_{x →-3}** 2x**^{3}** + 4x – 7**

**Q4. Find the value of lim **_{x →1}** 5x**^{2}** + 4x – 32**

## FAQs on Limits

### What is Limit in Calculus?

A limit in calculus is defined as a function that approaches a particular value as the independent variable of the function approaches a particular value.

### What is Limit Formula?

Limit formula is the formula that is used to calculate the limit of the given function. Let’s take a function y = f(x) and we take a point x = a then limit of function f(x) is defined as,

Iim_{x→0}f(x) = f(a)

### What is Limit of a Function Class 11?

A limit of a function f(x) is defined as a value, where the function reaches as the limit reaches some value.

### When can Limit Not Exist?

If the Left Hand Limit and the right hand limit of the function are not equal then we say that limit of the function does to exist, i.e. limit does not exit if,

LHL ≠ RHL

### How to Find Limit in Calculus?

Limit of the function is found by substituting the value of the function that the limit approaches if the limit exist. When the limit of the function does not exist then we first simplify the function and then find the limit of the function.

### What is the use of Limit?

The limit of the function is used for various purposes that are,

- Limits are used to find the limiting value of the function.
- Limits are used to find the derivative of the function.
- Limits are also used to define the integral value of the function, etc.

### What are Three Types of Limits?

The three types of limits are,

- Two-Sided Limits
- One-Sided Limits
- Infinite Limits