The document discusses various limits of functions, including special cases for polynomial, trigonometric, exponential, and logarithmic functions. Several examples and limits are provided to illustrate the concepts effectively. It aims to assist students in understanding and calculating limits in physics and mathematics.

1. Physics Helpline
L K Satapathy
Limits of Functions 1

2. Physics Helpline
L K Satapathy
Limits of Functions 1
1
lim
n n
n
x a
x a na
x a


 

0
sinlim 1
x
x
x

0
tanlim 1
x
x
x

0
1lim 1
x
x
e
x
  0
1lim log
x
x
a a
x
 
0
log(1 )
lim 1
x
x
x


0
sinlim
x
ax a
bx b

0
tanlim
x
ax a
bx b

0
sinlim
sinx
ax a
bx b

0
tanlim
tanx
ax a
bx b

Limits of Polynomial , Trigonometric , Exponential and Logarithmic Functions

3. Physics Helpline
L K Satapathy
Limits of Functions 1
Illustrations
7 7 7
7 1
1 1
1 1(1) lim lim 7 1 7
1 1x x
x x
x x

 
    
 
3 3 3
3 1
2 2
8 2(2) lim lim 3 2 12
2 2x x
x x
x x

 
    
 
3 33
3 1
3 3
( 3)27(3) lim lim 3 ( 3) 27
3 ( 3)x x
xx
x x

 
      
  
1
1
(4) lim lim lim
m m mm m n n
m n
n n nx a x a x a
x a ma mx a x a a
x a x a nx a na


  
    
 
6 6 6 16 6 3 3
6 3 3
3 3 3 1
6 6(5) lim lim lim 2
33x a x a x a
x a ax a x a a a
x a x ax a a


  
     
 

4. Physics Helpline
L K Satapathy
Limits of Functions 1
0 0 0
sin3 1 sin3 3 sin3 3 3(1) lim lim lim 1
5 5 5 3 5 5x x x
x x x
x x x  
      
0 0
0
0 0
sin3 sin3lim lim
sin3 3 33(2) lim
sin5 sin5 5 sin5 5lim lim
5
x x
x
x x
x x
x x x
x x x
x x
 

 
   
2
2 20 0 0 0
1 cos2 2sin sin sin(3) lim lim 2 lim lim 2
x x x x
x x x x
x xx x   
     
 2 20 0 0
cos3 cos7 2sin5 .sin2 sin5 sin2(4) lim lim 2 lim .
x x x
x x x x x x
x xx x  
   
0 0
sin5 sin 22 5 2 lim lim 20
5 2x x
x x
x x 
     
Illustrations

5. Physics Helpline
L K Satapathy
Limits of Functions 1
0 0
0
0 0
tan8 tan8lim lim
tan8 8 88(3) lim 4
sin 2 sin 2 2 sin 2 2lim lim
2
x x
x
x x
x x
x x x
x x x
x x
 

 
    
0 0
tan9 tan9(1) lim 3 lim 3 1 3
3 9x x
x x
x x 
    
0 0
0
0 0
tan6 tan6lim 6 lim
tan6 6 16(2) lim 2
tan3 tan3 tan3 3 1lim 3 lim
3
x x
x
x x
x x
x x x
x x x
x x
 

 

   

Illustrations

6. Physics Helpline
L K Satapathy
Limits of Functions 1
2 32 3 2 3
0 0 0 0
(3 1) (2 1)3 2 3 1 2 1(3) lim lim lim lim
x xx x x x
x x x xx x x x   
      
2 3
0 0
3 1 2 12 lim 3 lim 2log3 3log2
2 3
x x
x xx x 
      
 2 3 9log3 log2 log9 log8 log
8
    
4 4
2
0 0
3 1 3 1(1) lim 2 lim 2 log3 log3 log9
2 4
x x
x xx x 
      
0 0
log(1 5 ) log(1 5 )
(2) lim 5 lim 5 1 5
5x x
x x
x x 
 
    
Illustrations

7. Physics Helpline
L K Satapathy
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