This document contains examples and explanations of limits involving various functions. Some key points covered include: - Substitution can be used to evaluate limits, such as substituting 2 into -2x^3. - Left and right hand limits must agree for the overall limit to exist. - The limit of a piecewise function exists if the left and right limits are the same. - Graphs can help verify limit calculations and show discontinuities. - Special limits involving trigonometric and greatest integer functions are evaluated.

1. Chapter 12
Limits & An Introduction To Calculus
12.1 Limit of a Function
Proverbs 24:29. "Do not say, 'I will do to him just
as he has done to me; I will render to the man
according to his work.'"

2. 7 10 13 16 19
Consider : , , , , , ...
1 2 3 4 5

3. 7 10 13 16 19
Consider : , , , , , ...
1 2 3 4 5
3n + 4
an =
n

4. 7 10 13 16 19
Consider : , , , , , ...
1 2 3 4 5
3n + 4
an =
n
3n + 4
What is lim ?
n→∞ n

5. 7 10 13 16 19
Consider : , , , , , ...
1 2 3 4 5
3n + 4
an =
n
3n + 4
What is lim ?
n→∞ n
3n 4
+
n n
n
n

6. 7 10 13 16 19
Consider : , , , , , ...
1 2 3 4 5
3n + 4
an =
n
3n + 4
What is lim ?
n→∞ n
3n 4 4
+ 3+
n n n
n 1
n

7. 7 10 13 16 19
Consider : , , , , , ...
1 2 3 4 5
3n + 4
an =
n
3n + 4
What is lim ?
n→∞ n
3n 4 4
+ 3+
n n n 3
n 1
n
Verify by Graphing

8. Look at these graphically
3n + 4
What is lim ?
n→−∞ n

9. Look at these graphically
3n + 4
What is lim ?
n→−∞ n
It still is approaching 3

10. Look at these graphically
3n + 4
What is lim ?
n→−∞ n
It still is approaching 3
3n + 4
What is lim+ ?
n→0 n

11. Look at these graphically
3n + 4
What is lim ?
n→−∞ n
It still is approaching 3
3n + 4
What is lim+ ?
n→0 n
x approaches 0 from the positive side

12. Look at these graphically
3n + 4
What is lim ?
n→−∞ n
It still is approaching 3
3n + 4
What is lim+ ?
n→0 n
x approaches 0 from the positive side
It is ∞

13. 3n + 4
What is lim− ?
n→0 n

14. 3n + 4
What is lim− ?
n→0 n
−∞

15. 2
Consider : y1 = ( x − 2 ) + 3

16. 2
Consider : y1 = ( x − 2 ) + 3
what happens to:
2
lim+ ( x − 2 ) + 3
x→4
and
2
lim− ( x − 2 ) + 3
x→4

17. 2
Consider : y1 = ( x − 2 ) + 3
what happens to:
2
lim+ ( x − 2 ) + 3
x→4
and
2
lim− ( x − 2 ) + 3
x→4
in both cases, the limit approaches 7

18. If lim f ( x ) = lim f ( x ) = L
+ −
x→c x→c
then lim f ( x ) = L
x→c

19. If lim f ( x ) = lim f ( x ) = L
+ −
x→c x→c
then lim f ( x ) = L
x→c
The left-hand limit and the
right-hand limit must agree!

20. If lim f ( x ) = lim f ( x ) = L
+ −
x→c x→c
then lim f ( x ) = L
x→c
The left-hand limit and the
right-hand limit must agree!
also ... lim f ( x ) = f ( c )
x→c
as long as the limit exists!

21. If lim f ( x ) = lim f ( x ) = L
+ −
x→c x→c
then lim f ( x ) = L
x→c
The left-hand limit and the
right-hand limit must agree!
also ... lim f ( x ) = f ( c )
x→c
as long as the limit exists!
Substitution!

22. Find Each Limit
1. lim ( −2x 3
)
x→2

23. Find Each Limit
1. lim ( −2x 3
)
x→2
always try substitution first!

24. Find Each Limit
1. lim ( −2x 3
)
x→2
always try substitution first!
-16

25. Find Each Limit
1. lim ( −2x 3
)
x→2
always try substitution first!
-16
2. lim ( −2x 3
)
x→∞

26. Find Each Limit
1. lim ( −2x 3
)
x→2
always try substitution first!
-16
2. lim ( −2x 3
)
x→∞
−∞

27. 2
3. lim ( 2x − 3) − 4
x→3

28. 2
3. lim ( 2x − 3) − 4
x→3
5

29. 2
3. lim ( 2x − 3) − 4
x→3
5
⎛ 1 ⎞
4. lim ⎜ ⎟
x→∞ ⎝ x ⎠

30. 2
3. lim ( 2x − 3) − 4
x→3
5
⎛ 1 ⎞
4. lim ⎜ ⎟
x→∞ ⎝ x ⎠
0

31. 5. lim [ x ]
x→2

32. 5. lim [ x ] Greatest Integer
x→2 Function

33. 5. lim [ x ] Greatest Integer
x→2 Function

34. 5. lim [ x ] Greatest Integer
x→2 Function
lim
x→2 +
[ x] = 2

35. 5. lim [ x ] Greatest Integer
x→2 Function
lim
x→2 +
[ x] = 2
lim
x→2 −
[ x] = 1

36. 5. lim [ x ] Greatest Integer
x→2 Function
lim
x→2 +
[ x] = 2
lim
x→2 −
[ x] = 1
Because the left-hand and right-hand
limits do not agree, this limit does
not exist.

37. 2
x + 4x − 5
6. lim
x→1 x −1

38. 2
x + 4x − 5
6. lim
x→1 x −1
This is a line with a hole at (1,6).
The function does not exist at x=1,
but the limit does exist as the left-
hand and right-hand limits both
approach 6.

39. 2
x + 4x − 5
6. lim
x→1 x −1
This is a line with a hole at (1,6).
The function does not exist at x=1,
but the limit does exist as the left-
hand and right-hand limits both
approach 6.
Graph to verify
2
x + 4x − 5
lim =6
x→1 x −1

40. x ( cos x )
7. lim
x→0 sin x

41. x ( cos x )
7. lim
x→0 sin x
1

42. x ( cos x )
7. lim
x→0 sin x
1
x ( cos x )
8. lim
x→π sin x

43. x ( cos x )
7. lim
x→0 sin x
1
x ( cos x )
8. lim
x→π sin x
No limit exists

44. HW #7
We are what we repeatedly do.
Excellence, therefore, is not an act but a habit.
Aristotle