1. What is a Limit-an simple approach .pdf

INTRODUCTION TO LIMITS

1. What is a Limit
Ms.ChathuriRanawaka
(B.Sc.(Sp.)inMath.(Hons)(USJP−SL),M.Sc.(UoC−SL))
SriLankaTechnologyCampus

2. What is a limit ?

3. What is a limit ?
•One of the most basic and fundamental ideas of calculus
is limits.

4. What is a limit ?
•One of the most basic and fundamental ideas of calculus
is limits.
•Limits allow us to look at what happens in a very, very
small region around a point.

5. What is a limit ?
•One of the most basic and fundamental ideas of calculus
is limits.
•Limits allow us to look at what happens in a very, very
small region around a point.
•Two of the major formal definitions of calculus depend
on limits

6. The idea of Limit…….
Considerfindingtheareaofacircle:

7. The idea of Limit…….
Considerfindingtheareaofacircle:
n=3n=4 n=5…………………………… ........n=12

8. The idea of Limits…….
Considerfindingtheareaofacircle:
n=3n=4 n=5…………………………… ........n=12

9. The idea of Limits…….
Let�??????=??????
�
−??????+�.
Fillthetable.
?????? �(??????)
1.0
1.5
1.8
1.9
1.95
1.99
1.995
1.999

10. The idea of Limits…….
Let�??????=??????
�
−??????+�.
Fillthetable.
?????? �(??????)
1.0 2.000000
1.5 2.750000
1.8 3.440000
1.9 3.710000
1.95 3.852500
1.99 3.970100
1.995 3.985025
1.999 3.997001

11. The idea of Limits…….
Let�??????=??????
�
−??????+�.
Fillthetable.
?????? �(??????)
1.0 2.000000
1.5 2.750000
1.8 3.440000
1.9 3.710000
1.95 3.852500
1.99 3.970100
1.995 3.985025
1.999 3.997001
?????? �(??????)
3.0
2.5
2.2
2.1
2.05
2.01
2.005
2.001

12. The idea of Limits…….
Let�??????=??????
�
−??????+�.
Fillthetable.
?????? �(??????)
1.0 2.000000
1.5 2.750000
1.8 3.440000
1.9 3.710000
1.95 3.852500
1.99 3.970100
1.995 3.985025
1.999 3.997001
?????? �(??????)
3.0 8.000000
2.5 5.750000
2.2 4.640000
2.1 4.310000
2.05 4.152500
2.01 4.030100
2.005 4.015025
2.001 4.003001

13. The idea of Limits…….
Let�??????=??????
�
−??????+�.
Fillthetable.
Guesswhatis�??????as??????approaches2?
?????? �(??????)
1.0 2.000000
1.5 2.750000
1.8 3.440000
1.9 3.710000
1.95 3.852500
1.99 3.970100
1.995 3.985025
1.999 3.997001
?????? �(??????)
3.0 8.000000
2.5 5.750000
2.2 4.640000
2.1 4.310000
2.05 4.152500
2.01 4.030100
2.005 4.015025
2.001 4.003001

14. The idea of Limits…….
Let�??????=??????
�
−??????+�.
Guesswhatis�??????as??????approaches2?
“thelimitofthefunction�??????=??????
�
−??????+�as??????approaches2is
equalto4.”

15. The idea of Limits…….
Let�??????=??????
�
−??????+�.
Guesswhatis�??????as??????approaches2?
“thelimitofthefunction�??????=??????
�
−??????+�as??????approaches2is
equalto4.”

16. Definition of a Limit of a function
f(x)
f(x)
(x,f(x))
(x,f(x))
L
ax x x
No matter how xapproaches a,
f(x) approaches L.

17. Definition of a Limit of a function
Ifasxapproachesa(withoutactuallyattainingthevaluea),f(x)
approachesthenumberL,thenwesaythat
“Lis the limit of f(x) as xapproaches a”,
f(x)
f(x)
(x,f(x))
(x,f(x))
L
ax x x
No matter how xapproaches a,
f(x) approaches L.

18. Definition of a Limit of a function
Ifasxapproachesa(withoutactuallyattainingthevaluea),f(x)
approachesthenumberL,thenwesaythat
“Lis the limit of f(x) as xapproaches a”,
and write
f(x)
f(x)
(x,f(x))
(x,f(x))
L
ax x x
No matter how xapproaches a,
f(x) approaches L.lim ( )
xa
f x L



19. Left & Right Hand Limitslim ( )
xa
fx

 lim ( )
xa
fx

 2
-2
-5 5 0lim
3
0



x
x 0lim
3
0



x
x

20. Left & Right Hand Limits
Ex: Consider the graph of ????????????given below.

21. Left & Right Hand Limits
Ex: Consider the graph of ????????????given below.
Find left and right hand limits of
????????????as ??????approaches 0.

22. Left & Right Hand Limits
Ex: Consider the graph of ????????????given below.
Find left and right hand limits of
????????????as ??????approaches 0.
lim
??????→0
−
????????????=?
lim
??????→0
+
????????????=?

23. Left & Right Hand Limits
Ex: Consider the graph of ????????????given below.
Find left and right hand limits of
????????????as ??????approaches 0.
lim
??????→0
−
????????????=1
lim
??????→0
+
????????????=0

24. Limit of a function at a point
A function �??????has a limit as ??????approaches ??????

25. Limit of a function at a point
A function �??????has a limit as ??????approaches ??????if and only if it has
right and left hand limits

26. Limit of a function at a point
A function �??????has a limit as ??????approaches ??????if and only if it has
right and left hand limits and these one sided limits are equal:

27. Limit of a function at a point
A function �??????has a limit as ??????approaches ??????if and only if it has
right and left hand limits and these one sided limits are equal:
Ex:0
10
20
30
40
50
5 10 15 20
x
Does the limit exist for this
function as ??????approaches 15
?

28. Limit of a function at a point
A function �??????has a limit as ??????approaches ??????if and only if it has
right and left hand limits and these one sided limits are equal:
Ex:0
10
20
30
40
50
5 10 15 20
x
Does the limit exist for this
function as ??????approaches 15
?
lim
??????→15
−
????????????=
lim
??????→15
+
????????????=

29. Limit of a function at a point
A function �??????has a limit as ??????approaches ??????if and only if it has
right and left hand limits and these one sided limits are equal:
Ex:0
10
20
30
40
50
5 10 15 20
x
Does the limit exist for this
function as ??????approaches 15
?
lim
??????→15
−
????????????=20
lim
??????→15
+
????????????=36

30. Limit of a function at a point
A function �??????has a limit as ??????approaches ??????if and only if it has
right and left hand limits and these one sided limits are equal:
Ex:0
10
20
30
40
50
5 10 15 20
x
Does the limit exist for this
function as ??????approaches 15
?
lim
??????→15
−
????????????≠lim
??????→15
+
????????????

31. Limit of a function at a point
A function �??????has a limit as ??????approaches ??????if and only if it has
right and left hand limits and these one sided limits are equal:
Ex:0
10
20
30
40
50
5 10 15 20
x
Does the limit exist for this
function as ??????approaches 15
?
lim
??????→15
−
????????????≠lim
??????→15
+
????????????
So, lim
??????→15
????????????DNE

32. Limit of a function at a point
Ex: Find the limits of following functions as ??????approaches 1

33. Limit of a function at a point
Ex: Find the limits of following functions as ??????approaches 1
�??????=
(??????−�)(??????+�)
(??????−�)
g??????=
??????+�,??????≠�
�, ??????=�

34. Limit of a function at a point
Ex: Find the limits of following functions as ??????approaches 1
lim
??????→�
�(??????)=�
lim
??????→�
�(??????)=�
lim
??????→�
�(??????)=�

35. Limit of a function at a point
Ex: Find the limits of ????????????as ??????approaches 1, 2, 3, and 4.

36. Limit of a function at a point
Notice:
Limit is a number.
The limit can exist even when the function is not defined at a point or has a
value different from the limit.