The document discusses different types of limits in calculus and analysis. A limit describes the behavior of a function near a particular input value. There are several types of limits including one-sided limits as x approaches a from the left or right, direct substitution limits where the value is simply substituted, and limits involving techniques like factoring, rationalization, as x approaches infinity, trigonometric functions, or the number e.

1. Limit Of Function
And
Its Types

2. 
BY
Muhammad Adeel

3. 
 In mathematics, the limit of a function is a
fundamental concept in calculus and analysis
concerning the behavior of that function near a
particular input.
 Let f be a function defined on some open interval
containing a except possibly at a itself. Then, the
limit of f as x approaches a is L, written as
Limit Of Function
 lim ( )
x a
f x f a



4. 
 Limit of f(x) as x approaches a from the Left.
Lift Side Limit
 xf
ax 

lim

5. 
 Limit of f(x) as x approaches a from the Right.
Right Side Limit
 xf
ax 

lim

6. 
 The limit of f as x approaches -3 exists and it is equal
to 0 because
Existence OF Limit
 

xf
x 3
lim  3
lim 0
x
f x


2
9
( )
3
x
f x
x




7. 
 Limit By Direct Substitution
 Limit By Factoring
 Limit By Rationalization
 Limit At Infinity
 Trigonometric Limit
 Limit Involving Number e
Types Of Limit

8.  These are easiest problems. In these problems you
only need to substitute the value to which the
independent value is approaching.
Direct Substitution

9. 
 We use our algebraic skills to simplify the expression
Factoring

10.  In these limits we apply an algebraic technique
called rationalization
Rationalization

11.  In these limits the independent variable is
approaching infinity
At Infinity

12.  In most limits that involve trigonometric functions
you must apply the fundamental limit:
Trigonometric

13. 
 This types of question have e form
Involving Number e

14. 