Limit is a topic of calculus. They are **mathematical construct** we can use to describe the behavior of a function near a point. We can use limits to find the derivative, continuity and integral of a function.

**What is Limit?**

The limit of the function f(x) is the value in which x approaches to some number.

If f is the real valued function and a is the real number then the above formula is read as the limit of f of x as x approaches to a equal to b.

**Limits are of three types**

Left hand limit

Right hand limit

Two-sided limit

**Left hand limit**

When a function has a limit x approaches to a from the left it is known as left hand limit.

lim┬(x→a^- )〖f(x)〗 = L

Example:

Evaluatelim┬(x→2^- )〖((x^2-3x+4)/(x-1))〗

Solution:

Step 1:Put the limit in the function

lim┬(x→2^- )〖((x^2-3x+4)/(x-1))〗=((3^2-3(3)+4)/(5-3(3)))

Step 2:Solve the equation,

= ((9-9+4)/(5-9))

= ((0+4)/(-4))= (4/(-4))= -1

Step 3:Write the function with answer,

lim┬(x→2^- )((x^2-3x+4)/(x-1))= -1

Graph

**Right hand limit**

When a function has a limit x approaches to a from the right it is known as right hand limit.

lim┬(x→a^+ )〖f(x)〗 = L

Example:

Evaluate lim┬(x→2^+ )〖((x^2+2)/(x-1))〗

Solution:

lim┬(x→2^+ )〖((x^2+2)/(x-1))〗 = (2^2+2)/(2-1)

= (4+2)/(2-1) = 6

Graph

**Two-sided limit**

A function f has limit L as x→a if and only if

lim┬(x→a^- )〖f(x)〗= L =lim┬(x→a^+ )〖f(x)〗

**Example 1:**

Evaluate lim┬(x→2)〖 ((x^2+2)/(x-1))〗

**Solution:**

Put the above equation in limit calculator to avoid the manual step-by-step calculation.

**Step 1: Apply the limit x➜2**

lim┬(x→2)((x^2+2)/(x-1))= (2^2+2)/(2-1)

**Step 2:Solve the equation**

= (4+2)/(2-1) = 6

**Step 3:Write the equation with result**

lim┬(x→2)〖((x^2+2)/(x-1))〗 = 6

**Example 2:**

Limx→3 ((x^2-3x+4)/(5-3x))

**Solution:**

Step 1:Put the limit in the function

Limx→3 ((x^2-3x+4)/(5-3x)) =((3^2-3(3)+4)/(5-3(3)))

**Step 2:Solve the equation,**

= ((9-9+4)/(5-9))

= ((0+4)/(-4))

= (4/(-4))

= -1

**Step 3:Write the function with answer,**

Limx→3 ((x^2-3x+4)/(5-3x)) = -1

Graph

**Notation of limit:**

We denote the limit as

lim┬(x→a)〖(f(x))〗 = L

Where f is a function, a is the limit of that function and L is the output of the function after applying the limit.

**Rules of limit.**

**Constant Rule**

When we apply limit on a constant it gives the result constant itself as limits only apply on variables not on constant.

lim┬(x→a)(k)= k

**Example:**

Find the limit of lim┬(x→2)4

**Solution:**

By constant rule limit of a constant is the constant itself

lim┬(x→2)4 = 4

**Constant times a function**

When a function is multiplied by a constant, we use this rule. According to this rule we bring the constant outside the limit,

lim┬(x→a)(kf(x) )= k lim┬(x→a)〖(f(x))〗

**Example:**

Find the limit oflim┬(x→2)〖(8x^2)〗?

**Solution:**

By constant times a function

lim┬(x→2)(8x^2 )= 〖8 lim┬(x→2)〗〖(x^2)〗

Now apply limit

lim┬(x→2)(8x^2 )= 8(22)

= 8(4) = 32

**Sum Rule**

When limit is applied on the sum of two functions, we use this rule. According to this rule we apply limits on both functions separately.

lim┬(x→a)(f(x)+g(x) )= lim┬(x→a)〖(f(x))〗+lim┬(x→a)〖(g(x))〗

**Example:**

Find the limit of lim┬(x→2)〖(x^2+x^3)〗

**Solution:**

By using sum rule

lim┬(x→2)〖(x^2+x^3)〗= lim┬(x→2)〖(x^2)〗+ lim┬(x→2)〖(x^3)〗

Now apply the limits

lim┬(x→2)〖(x^2+x^3)〗= (22) + (23)

= 4 + 8

= 12

**Difference Rule**

When limit is applied on the difference of two functions, we use this rule. According to this rule we apply limits on both functions separately.

lim┬(x→a)〖(f(x)-g(x))〗= lim┬(x→a)〖(f(x))〗-lim┬(x→a)〖(g(x))〗

**Example:**

Find the limit of lim┬(x→2)〖(x^2-x^3)〗

**Solution:**

By using Difference rule

lim┬(x→2)〖(x^2-x^3)〗= lim┬(x→2)〖(x^2)〗- lim┬(x→2)〖(x^3)〗

Now apply the limits

lim┬(x→2)〖(x^2-x^3)〗= (22) – (23)

= 4 – 8

= -4

**Product Rule**

When limit is applied on the product of two functions, we use this rule. According to this rule we apply limits on both functions separately.

lim┬(x→a)(f(x)*g(x) )= lim┬(x→a)〖(f(x))〗*lim┬(x→a)〖(g(x))〗

**Example:**

Find the limit of lim┬(x→2)〖(x^2*x^3)〗

**Solution:**

By using Product rule

lim┬(x→2)〖(x^2*x^3)〗= lim┬(x→2)〖(x^2)〗* lim┬(x→2)〖(x^3)〗

Now apply the limits

lim┬(x→2)〖(x^2*x^3)〗= (22) *(23)

= 4 * 8

= 32

**Quotient Rule**

When limit is applied on the division of two functions, we use this rule. According to this rule we apply limits on both functions separately.

lim┬(x→a)(f(x)/g(x) )= lim┬(x→a)(f(x) )/lim┬(x→a)〖(g(x))〗

Where g(x) is not equal to zero

**Example:**

Find the limit of lim┬(x→2)〖(x^2/X^3 )〗

**Solution:**

By using quotient rule

lim┬(x→2)〖(x^2/x^3 )〗= (lim┬(x→2) x^2)/(lim┬(x→2) x^3 )

Now apply the limits

lim┬(x→2)〖(x^2/x^3 )〗= 2^2/2^3

=4/8

= 1/2

Function raised to an exponent

When a function is given with the power of whole function, we use this rule.

lim┬(x→a)〖[f(x) ]^n 〗= [lim┬(x→a) f(x) ]^n

**Example:**

Evaluate lim┬(x→2)〖[3x+1]^5 〗

**Solution:**

By function raised to an exponent rule

lim┬(x→2)〖[3x+1]^5 〗 = 〖[lim┬(x→2) (3x+1)]^5 〗

= 〖[lim┬(x→2) (3x)+〖lim(〗┬(x→2) 1)]^5 〗

= 〖[〖3lim〗┬(x→2) (x)+lim┬(x→2) (1)]^5 〗

= 〖[3(2)+1]^5 〗

= 〖[6+1]^5 〗

= 〖[7]^5 〗

= 16,805

**How to find limits?**

We can find the limits by using the general rules of limits. However, an online **limit calculator with steps** can ease up your limits calculations by providing you the step by step calculations. You can use those steps to complete your assignments or prepare for exams.

Limits are used to find derivatives, integrals and continuity of functions.

Let us learn how to find limits by using examples

**Example 1:**

Evaluate lim┬(x→3)〖(2x+3)〗

**Solution:**

lim┬(x→3)〖(2x+3)〗 = lim┬(x→3)〖(2x)〗 + lim┬(x→3)〖(3)〗 (By sum rule)

= 〖2 lim〗┬(x→3)〖(x)〗 + 3 (By constant rules)

= 2(3) + 3

= 6 + 3 = 9

**Example 2:**

Evaluate lim┬(x→3)〖(x^2-6x+8)/(x-2)〗

**Solution:**

lim┬(x→3)〖(x^2-6x+8)/(x-2)〗 = lim┬(x→3)〖((x-2)(x-4))/(x-2)〗 (By factorization)

= lim┬(x→3)〖((x-4))/1〗 (By cancelling the common)

= 3 – 4

= -1

**Some general results**

lim┬(x→a)〖(x^n-a^n)/(x-a)〗 = n a(n-1), for all real values of n

lim┬(x→0)〖(sin(x))/x〗 = 1

〖lim〗┬(x→0)〖(tan(x))/x〗 = 1

lim┬(x→0)〖(1-cos(x))/x〗 = 0

lim┬(x→0) 〖 cos〗(x)=1

lim┬(x→0) 〖 e〗^x = 1

lim┬(x→0)〖(e^x-1)/x〗 = 1

lim┬(x→∞)〖(1+1/x)〗 = e

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