1. Limits & Continuity

2. Limits & Continuity
A limit describes the behaviour of functions.

3. Limits & Continuity
A limit describes the behaviour of functions.
 lim :
x a
f x


4. Limits & Continuity
A limit describes the behaviour of functions.
 lim :
x a
f x

as the x value approaches a, what value does f(x) approach?

5. Limits & Continuity
A limit describes the behaviour of functions.
 lim :
x a
f x

as the x value approaches a, what value does f(x) approach?
 lim :
x a
f x


6. Limits & Continuity
A limit describes the behaviour of functions.
 lim :
x a
f x

as the x value approaches a, what value does f(x) approach?
 lim :
x a
f x

as the x value approaches a from the negative side,
what value does f(x) approach?

7. Limits & Continuity
A limit describes the behaviour of functions.
 lim :
x a
f x

as the x value approaches a, what value does f(x) approach?
 lim :
x a
f x

as the x value approaches a from the negative side,
what value does f(x) approach?
y
x
1
1
1y x 

8. Limits & Continuity
A limit describes the behaviour of functions.
 lim :
x a
f x

as the x value approaches a, what value does f(x) approach?
 lim :
x a
f x

as the x value approaches a from the negative side,
what value does f(x) approach?
y
x
1
1
1y x 
1
lim 1 0
x
x

 

9. Limits & Continuity
A limit describes the behaviour of functions.
 lim :
x a
f x

as the x value approaches a, what value does f(x) approach?
 lim :
x a
f x

as the x value approaches a from the negative side,
what value does f(x) approach?
y
x
1
1
1y x 
1
lim 1 0
x
x

 
y
x
4
6
 y f x

10. Limits & Continuity
A limit describes the behaviour of functions.
 lim :
x a
f x

as the x value approaches a, what value does f(x) approach?
 lim :
x a
f x

as the x value approaches a from the negative side,
what value does f(x) approach?
y
x
1
1
1y x 
1
lim 1 0
x
x

 
y
x
4
6
 y f x
 0
lim 4
x
f x



11. Limits & Continuity
A limit describes the behaviour of functions.
 lim :
x a
f x

as the x value approaches a, what value does f(x) approach?
 lim :
x a
f x

as the x value approaches a from the negative side,
what value does f(x) approach?
y
x
1
1
1y x 
1
lim 1 0
x
x

 
y
x
4
6
 y f x
 0
lim 4
x
f x


 lim :
x a
f x


12. Limits & Continuity
A limit describes the behaviour of functions.
 lim :
x a
f x

as the x value approaches a, what value does f(x) approach?
 lim :
x a
f x

as the x value approaches a from the negative side,
what value does f(x) approach?
y
x
1
1
1y x 
1
lim 1 0
x
x

 
y
x
4
6
 y f x
 0
lim 4
x
f x


 lim :
x a
f x

as the x value approaches a from the positive side,
what value does f(x) approach?

13. Limits & Continuity
A limit describes the behaviour of functions.
 lim :
x a
f x

as the x value approaches a, what value does f(x) approach?
 lim :
x a
f x

as the x value approaches a from the negative side,
what value does f(x) approach?
y
x
1
1
1y x 
1
lim 1 0
x
x

 
y
x
4
6
 y f x
 0
lim 4
x
f x


 lim :
x a
f x

as the x value approaches a from the positive side,
what value does f(x) approach?
1
lim 1 0
x
x

 

14. Limits & Continuity
A limit describes the behaviour of functions.
 lim :
x a
f x

as the x value approaches a, what value does f(x) approach?
 lim :
x a
f x

as the x value approaches a from the negative side,
what value does f(x) approach?
y
x
1
1
1y x 
1
lim 1 0
x
x

 
y
x
4
6
 y f x
 0
lim 4
x
f x


 lim :
x a
f x

as the x value approaches a from the positive side,
what value does f(x) approach?
1
lim 1 0
x
x

   0
lim 6
x
f x



15. Limits & Continuity
A limit describes the behaviour of functions.
 lim :
x a
f x

as the x value approaches a, what value does f(x) approach?
 lim :
x a
f x

as the x value approaches a from the negative side,
what value does f(x) approach?
y
x
1
1
1y x 
1
lim 1 0
x
x

 
y
x
4
6
 y f x
 0
lim 4
x
f x


 lim :
x a
f x

as the x value approaches a from the positive side,
what value does f(x) approach?
1
lim 1 0
x
x

   0
lim 6
x
f x


     If lim lim , then is continuous at
x a x a
f x f x f x x a 
 
 

16. Finding Limits

17. Finding Limits(1) Direct Substitution

18. Finding Limits(1) Direct Substitution
5
e.g. lim 7
x
x



19. Finding Limits(1) Direct Substitution
5
e.g. lim 7
x
x

 5 7
12
 


20. Finding Limits(1) Direct Substitution
5
e.g. lim 7
x
x

 5 7
12
 

(2) Factorise and Cancel

21. Finding Limits(1) Direct Substitution
5
e.g. lim 7
x
x

 5 7
12
 

(2) Factorise and Cancel
2
3
9
e.g. lim
3x
x
x



22. Finding Limits(1) Direct Substitution
5
e.g. lim 7
x
x

 5 7
12
 

(2) Factorise and Cancel
2
3
9
e.g. lim
3x
x
x


  
 3
3 3
lim
3x
x x
x
 



23. Finding Limits(1) Direct Substitution
5
e.g. lim 7
x
x

 5 7
12
 

(2) Factorise and Cancel
2
3
9
e.g. lim
3x
x
x


  
 3
3 3
lim
3x
x x
x
 


 3
lim 3
x
x

 

24. Finding Limits(1) Direct Substitution
5
e.g. lim 7
x
x

 5 7
12
 

(2) Factorise and Cancel
2
3
9
e.g. lim
3x
x
x


  
 3
3 3
lim
3x
x x
x
 


 3
lim 3
x
x

 
3 3
6
 


25. (3) Special Limit

26. (3) Special Limit
1
lim 0
x x


27. (3) Special Limit
1
lim 0
x x

 
3 2
3
3 2 1
e.g. lim
4 1x
x x x
i
x
  


28. (3) Special Limit
1
lim 0
x x

 
3 2
3
3 2 1
e.g. lim
4 1x
x x x
i
x
  

3 2
3 3 3 3
3
3 3
3 2 1
lim
4 1x
x x x
x x x x
x
x x

  



29. (3) Special Limit
1
lim 0
x x

 
3 2
3
3 2 1
e.g. lim
4 1x
x x x
i
x
  

3 2
3 3 3 3
3
3 3
3 2 1
lim
4 1x
x x x
x x x x
x
x x

  


2 3
3
3 2 1
1
lim
1
4
x
x x x
x

  



30. (3) Special Limit
1
lim 0
x x

 
3 2
3
3 2 1
e.g. lim
4 1x
x x x
i
x
  

3 2
3 3 3 3
3
3 3
3 2 1
lim
4 1x
x x x
x x x x
x
x x

  


2 3
3
3 2 1
1
lim
1
4
x
x x x
x

  


1
4


31. (3) Special Limit
1
lim 0
x x

 
3 2
3
3 2 1
e.g. lim
4 1x
x x x
i
x
  

3 2
3 3 3 3
3
3 3
3 2 1
lim
4 1x
x x x
x x x x
x
x x

  


2 3
3
3 2 1
1
lim
1
4
x
x x x
x

  


1
4

 
2
3
4
lim
1x
x x
ii
x



32. (3) Special Limit
1
lim 0
x x

 
3 2
3
3 2 1
e.g. lim
4 1x
x x x
i
x
  

3 2
3 3 3 3
3
3 3
3 2 1
lim
4 1x
x x x
x x x x
x
x x

  


2 3
3
3 2 1
1
lim
1
4
x
x x x
x

  


1
4

 
2
3
4
lim
1x
x x
ii
x


0
1
0



33. (3) Special Limit
1
lim 0
x x

 
3 2
3
3 2 1
e.g. lim
4 1x
x x x
i
x
  

3 2
3 3 3 3
3
3 3
3 2 1
lim
4 1x
x x x
x x x x
x
x x

  


2 3
3
3 2 1
1
lim
1
4
x
x x x
x

  


1
4

 
2
3
4
lim
1x
x x
ii
x


0
1
0


 
7 6 2
7
lim
3 974x
x x x
iii
x x
 
 

34. (3) Special Limit
1
lim 0
x x

 
3 2
3
3 2 1
e.g. lim
4 1x
x x x
i
x
  

3 2
3 3 3 3
3
3 3
3 2 1
lim
4 1x
x x x
x x x x
x
x x

  


2 3
3
3 2 1
1
lim
1
4
x
x x x
x

  


1
4

 
2
3
4
lim
1x
x x
ii
x


0
1
0


 
7 6 2
7
lim
3 974x
x x x
iii
x x
 
 
1
3


35.  
3
2
2
lim
1x
x
iv
x



36.  
3
2
2
lim
1x
x
iv
x


1
0

 

37.  
3
2
2
lim
1x
x
iv
x


1
0

 
 
  
  
3 2
Find the horizontal asymptote of
1 1
x x
v y
x x
 

 

38.  
3
2
2
lim
1x
x
iv
x


1
0

 
 
  
  
3 2
Find the horizontal asymptote of
1 1
x x
v y
x x
 

 
  
  
3 2
lim
1 1x
x x
x x
 
 

39.  
3
2
2
lim
1x
x
iv
x


1
0

 
 
  
  
3 2
Find the horizontal asymptote of
1 1
x x
v y
x x
 

 
  
  
3 2
lim
1 1x
x x
x x
 
 
2
2
6
lim
1x
x x
x
 



40.  
3
2
2
lim
1x
x
iv
x


1
0

 
 
  
  
3 2
Find the horizontal asymptote of
1 1
x x
v y
x x
 

 
  
  
3 2
lim
1 1x
x x
x x
 
 
2
2
6
lim
1x
x x
x
 


1
1
1



41.  
3
2
2
lim
1x
x
iv
x


1
0

 
 
  
  
3 2
Find the horizontal asymptote of
1 1
x x
v y
x x
 

 
  
  
3 2
lim
1 1x
x x
x x
 
 
2
2
6
lim
1x
x x
x
 


1
1
1


horizontal asymptote is 1y 

42.  
3
2
2
lim
1x
x
iv
x


1
0

 
 
  
  
3 2
Find the horizontal asymptote of
1 1
x x
v y
x x
 

 
  
  
3 2
lim
1 1x
x x
x x
 
 
2
2
6
lim
1x
x x
x
 


1
1
1


horizontal asymptote is 1y  Exercise 7I; 1a, 2ace, 3ac,
4a, 5ad, 8a, 9ab, 10a