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How are limits interpreted in calculus and how to find them?

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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