Limit of functions

1. One-sided limit, limit, existence of limit, limit at
infinity, infinite limit

2. One-sided limit
 xf
ax 

lim
Limit of f(x) as x approaches a from the left.

3. One-sided limit
 xf
ax 

lim
Limit of f(x) as x approaches a from the right.

4. One-sided limit
Consider the function defined by
 ?lim
3
xf
x 

 ?lim
3
xf
x 


5. One-sided limit
Where does f(x) point to as x approaches -3
from the left?
 ?lim
3
xf
x 

 ?lim
3
xf
x 

Where does f(x) point to as x approaches -3
from the right?

6. One-sided limit
 ?lim
3
xf
x 

 ?lim
3
xf
x 


7. One-sided limit
    
3where
3
3
33
3
92








x
x
x
xx
x
x
xf

8. One-sided limit
x  
3
92



x
x
xf x  
3
92



x
x
xf
-3.1 -6.1 -2.9 -5.9
-3.01 -6.01 -2.99 -5.99
-3.001 -6.001 -2.999 -5.999
-3.0001 -6.0001 -2.9999 -5.9999
Table 2.1 Some numerical computations close to -3
from the left and right

9. One-sided limit
  6lim
3


xf
x
  6lim
3


xf
x

10. Existence of Limit
Given a function defined by
The limit of f as x approaches -3 exists and it is
equal to -6 because
 

xf
x 3
lim   6lim
3


xf
x

11. Existence of Limit
  6lim
3


xf
x

12. One-sided limit
Functions whose limit at a does not exist.
Example 2.1
Evaluate lim
𝑥→1
𝑓 𝑥 , does it exist?Does the limit exists as x approaches 1?

13. One-sided limit
Conclusion: the limit does not exist
Use the criteria to prove existence of limit
at x = 1

14. Existence of Limit

15. Existence of Limit

16. Existence of limit

17. Existence of Limit
The limits do not point to a
specific real number.
Conclusion: limit does not
exist as x approaches 3
x = 3 is the vertical
asymptote

18. Existence of Limit
L is a specific real number

19. Limit Theorems

20. Limit at infinity
Carrying capacity

21. Limit at infinity
As x approaches
positive infinity or
negative infinity, the
quotient approaches
zero.

22. Limit at infinityEvaluate lim
𝑥→∞
𝑓 𝑥 where
Solution

23. Limit at Infinity and Horizontal
Asymptote
Obtain the horizontal asymptote of
Solution

24. Limit at infinity and horizontal
asymptote
Horizontal
Asymptote 𝑦 = −1

25. Limit at infinity and asymptotes
(horizontal and oblique)

26. Limit at infinity and asymptotes
(horizontal and oblique)
Examples

27. Limit at infinity and asymptotes
(horizontal and oblique)
Perform long division on

28. Limit of some trigonometric
functions
Theorems
lim
𝑥→0
sin 𝑥
𝑥
= 1 lim
𝑥→0
cos 𝑥 − 1
𝑥
= 0

29. Limit of trigonometric functions

30. Limit of trigonometric functions

31. Limit of trigonometric functions

32. Limit of trigonometric functions