Formulae and Illustrations of Limits of Polynomial , Trigonometric , Logarithmic and Exponential Functions

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
For More details:
www.physics-helpline.com
Subscribe our channel:
youtube.com/physics-helpline
Follow us on Facebook and Twitter:
facebook.com/physics-helpline
twitter.com/physics-helpline