Transcript Lesson 6-2

Lesson 2-2 part 2
The Limit of a Function
Objectives
• Determine and Understand one-sided limits
• Determine and Understand two-sided limits
Vocabulary
• Limit (two sided) – as x approaches a value a, f(x) approaches a
value L
• Left-hand (side) Limit – as x approaches a value a from the
negative side, f(x) approaches a value L
• Right-hand (side) Limit – as x approaches a value a from the
positive side, f(x) approaches a value L
• DNE – does not exist (either a limit increase/decreases without
bound or the two one-sided limits are not equal)
• Infinity – increases (+∞) without bound or decreases (-∞)
without bound [NOT a number!!]
• Vertical Asymptote – at x = a because a limit as x approaches a
either increases or decreases without bound
Limits
When we look at the limit below, we examine the f(x) values as x gets very
close to a:
read: the limit of f(x), as x approaches a, equals L
lim f(x) = L
xa
One-Sided Limits:
Left-hand limit (as x approaches a from the left side – smaller)
lim f(x) = L
xa-
f(x) = L
RIght-hand limit (as x approaches a from the right side – larger) lim
xa
+
The two-sided limit (first one shown) = L if and only if both one-sided limits = L
lim f(x) = L
xa
if and only if
lim f(x) = L
xa-
and lim f(x) = L
xa+
Vertical Asymptotes:
The line x = a is called a vertical asymptote of y = f(x) if at least one of the
following is true:
lim f(x) = ∞
xa
lim f(x) = -∞
xa
lim f(x) = ∞
xa-
lim f(x) = -∞
xa-
lim f(x) = ∞
xa+
lim f(x) = -∞
xa+
Limits Using Graphs
Usually a reasonable
guess would be:
y
One Sided Limits
Limit from right:
lim f(x) = f(a)
lim f(x) = 5
xa
x10+
(this will be true for
continuous functions)
ex:
Limit from left:
lim f(x) = 3
lim f(x) = 2
x10-
x2
but,
Since the two onesided limits are not
equal, then
lim f(x) = 7
x5
(not f(5) = 1)
and lim f(x) = DNE
x
x16
(DNE = does not exist)
2
5
10
15
When we look at the limit below, we examine
the f(x) values as x gets very close to a:
lim f(x)
xa
lim f(x) = DNE
x10
lim f  x  
x5
Example 1
Answer each using the graph to the right
a.
Lim f(x) =
1
x→ -2-
b.
Lim f(x) =
0
x→ -2+
c.
Lim f(x) =
DNE
x→ -2
d.
Lim f(x) =
3
x→ 2-
e.
Lim f(x) =
0
x→ 2+
f.
Lim f(x) =
DNE
x→ 2
g.
Lim f(x) =
1
x→ 0-
h.
Lim f(x) =
1
x→ 0+
i.
Lim f(x) =
x→ 0
1
Example 2
Find
3-x
Lim -----------|x – 3|
x→ 3-
=
1
Example 3
Find:
a. Lim f(x)
x→ 2
if f(x) =
3x + 1
x<2
8
x=2
x² + 3
x>2
Lim f(x) = 7
x→ 2
Example 4
Always, Sometimes or Never True:
a. If
Lim f(x)
x→ 2
does not exist, then
Lim f(x)
x→ 2+
does not exist.
Sometimes --- if a two-sided limit is DNE,
then a one-sided limit might be DNE
b. If
Lim f(x)
x→ 2+
Always
does not exist, then Lim f(x)
x→ 2
does not exist.
--- if a one-sided limit is DNE,
then the two-sided limit must be DNE
lim f  x  
x5
Example 5
Answer each using the graph to the right
a.
Lim f(x) =
DNE (+ )
x→ -2-
b.
Lim f(x) =
DNE (+ )
x→ -2+
c.
Lim f(x) =
DNE (+ )
x→ -2
d.
Lim f(x) =
DNE (+ )
x→ 3-
e.
Lim f(x) =
0
x→ 3+
f.
Lim f(x) =
DNE only
x→ 3
g.
Lim f(x) =
x→ 0-
h.
Lim f(x) =
x→ 0+
i.
Lim f(x) =
x→ 0
DNE (- )
DNE (+ )
DNE only
Example 6
True/False: If
Lim f(x) = 
x→ a
and Lim g(x) = 
x→ a
,
then Lim [f(x) – g(x)] = 0 .
x→ a
False
Summary & Homework
• Summary:
– Try to find the limit via direct substitution
– Use algebra to simplify into useable form
– Graph the function
• Homework: pg 102-104: 12, 19, 21, 23,
24, 27 ;