Calculus: Chapter 1: Limit of the Function
Chapter 1: Limit of the Function
The limit is the first character
of any function which tells about the nature of the function. 
. The
limit represents the value that 
approaches as 
gets
arbitrarily close to 
from both sides
(left and right). Limit is the foundation of many properties of function
like continuity, differentiability etc.
Limits Definition
When examining a real-valued
function and a real
number "," the limit is typically expressed as
follows:
"The limit of as x approaches
c equals 
." The notation "lim" denotes the
limit, while the arrow signifies that the function converges to
the limit as x approaches
.
Properties of Limit:
1. Linearity: The limit of a sum or difference of functions is the sum or difference of their limits.
Sum
Rule: 
Example
: 
Difference Rule:
Example
: 
2. Product Rule: The limit of a product of functions is the product of their limits.
Example
: 
3. Quotient Rule: The limit of a quotient of functions is the quotient of their limits, provided the limit of the denominator is not zero.
Example
: 
4. Constant Rule: The limit of a constant times a function is the constant times the limit of the function.
Example
: 
5. Power Rule: The limit of the power of a function is the power of the limit of the function. If m and n are integers than Provided is a real number.
Provided 
is a real
number.
The existence of Limit: The existence of the limit of a function hinge on the equality of its left-hand and right-hand limits.
Limits of Trigonometric Functions
1. 
2. 
3. 
4. 
Limits of Exponential Functions
In particular 
.
In general, if 
, then 
Indeterminant form: The form of
function whose values can't be determined analytically. for example: 
etc.
Solved Examples on Limits
Example 1:
Evaluate 
Solution:
Example 2:
Evaluate
Solution:
Example 3:
Evaluate
Solution:
Example 4:
Evaluate
Solution:
Example 5:
Evaluate
Solution:
Example 6:
Evaluate
Solution:
No limit exists
Example 7:
Evaluate
Solution:
Example 8:
Evaluate
Solution:
Limit does not exist.
Example 9: Given function
Evaluate the following limits,
i)
ii)
Solution:
i)
We know that 
therefore we take 
ii)
We take 
because here it is 
Example 10: Find the limit of the function 
as 
approaches 2.
Solution: To find the limit as approaches 2 , we can try direct substitution.
At 
Here we get an indeterminate from 
indicating that we need to simplify the
expression further we can factorize the numerator.
Now, we can cancel out
the common factor of 
.
Now, let's find the limit
as a approaches 
So, the limit of the function 
is 4 .
Example 11:
Find the limit of the Function 
as approaches 2
Solution:
We have given
At 
Applying factorization:
Cancelling the common
factor:
Taking the limit as x
approaches 2 :
Example
12: Find the limit of the function 
as approaches 0
Solution: We have given
At 
Using trigonometric limits:
Example
13: Find the limit of the function 
as approaches 2.
Solution:
We have given
At 
,
Using factorization: 
Cancelling the common factor:
Taking the limit as x
approaches 2 :
So, 
Example 14:
Find the limit of the function 
as approaches 
.
Solution:
At 
Rationalize the numerator:
Taking
the limit as x approaches 0 :
Example
15: Find the limit of the function 
as approaches 
Solution:
We have given
At 
,
Using trigonometric
identities:
So, 
Example 16: Find
the limit of the function 
.
Solution:
Assignment Questions:
1.
Evaluate 
2.
Evaluate 
3.
Evaluate 
4.
Evaluate 
5. Evaluate
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