1.4--Continuity & One-Sided Limits
Find the limit (if it exists):
1)
3x - 20
lim
7+
x
2)
5x + 10
lim
x
-
9
3)
lim
x
2x - 14
4
x - 9
x - 2
-
x - 4
3x - 7
4)
lim
x
f(x), where f(x) =
5
4
2x2 - 8
5x - 4
, x ≤ 5
, x > 5
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) =
1.4--Continuity & One-Sided Limits
3 types of discontinuity:
1)
point
point
jump
infinite
1.4--Continuity & One-Sided Limits
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)
15)
16)
y =
y =
y =
x2 - 11x - 26
x2 + 3x - 40
x2 - 7x + 12
x3 - 9x
[x]
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:
17)
18)
f(x) =
f(x) =
3x - 9
x - 3
4x - 8 , x ≤ 5
x + 1
x2 - 19 , x > 5
19)
Find the value of k that will make the function continuous:
3x + 1
f(x) =
6
2x2 + 10
7k - 5
, x ≤ 4
, x > 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
20)
Verify that the Intermediate Value Theorem applies to
the indicated interval and find the value of c guaranteed
by the theorem:
f(x) = x2 - 7x + 2,
[0,9],
f(c) = 6