1.4--Continuity & One-Sided Limits
Find the limit (if it exists):
1)
2)
3)
x
x
x
4x - 12
lim
3
+
2x2 - 18
2x + 1
lim
5
-
9
-
x - 5
x - 3
lim
x - 9
5x - 3
4)
x
lim
2
f(x), where f(x) =
4
2x2 - 1
3x - 2
, x≤ 2
, x> 2
1
1.4--Continuity & One-Sided Limits
5)
Does f(-5) exist?
6)
Does lim f(x) exist?
7)
Does lim f(x) exist?
8)
Does f(-2) exist?
9)
Does lim f(x) exist?
10)
Does lim f(x) = f(-2)?
11)
lim f(x) =
12)
lim f(x) =
13)
lim f(x) =
2
1.4--Continuity & One-Sided Limits
3 types of discontinuity:
1)
point
2)
3)
jump
infinite
In order for f(x) to be continuous at x = c,
all 3 of these must be true:
(A)
(B)
(C)
f(c) must exist. (No holes or asymptotes)
lim f(c) exists.
x
c
lim f(x) = f(c). (What you expect is what you
c
get upon reaching x = c).
x
Find any points of discontinuity. Tell which type of
discontinuity it is (point, jump, or infinite):
14)
16)
y =
y =
3
x-2
[x]
15)
17)
y =
y =
x2 - 3x - 18
2x + 6
x2 - 9x + 20
x2 - 4x
3
1.4--Continuity & One-Sided Limits
Find the x-values (if any) at which f(x) is not continuous.
State whether the discontinuity is removable or not:
18)
19)
f(x) =
f(x) =
2x + 8
x + 4
5x - 8 , x ≤ 4
x
x2 - 13 , x > 4
20)
Find the value of k that will make the function continuous:
5x - 3
f(x) =
4
2x2 + 9
3k - 2
, x≤ 6
, x> 6
k = 14/3
4
1.4--Continuity & One-Sided Limits
The Intermediate Value Theorem
If f is continuous on the closed interval [a,b] and k is any
number between f(a) and f(b), then there is at least one
number c in [a,b] such that f(c) = k.
f(b)
f(a)
b
a
1.4--Continuity & One-Sided Limits
21)
Use the Intermediate Value Theorem & graphing calculator
to approximate the zero of the function on [0,1] accurate
to 4 decimal places:
f(x) = 3x2 + 8x - 5
5
1.4--Continuity & One-Sided Limits
22)
Verify that the Intermediate Value Theorem applies to
the indicated interval and find the value of c guaranteed
by the theorem:
f(x) = 2x2 - 11x + 5,
[1,6],
f(c) = 4
6