MA108 CALCULUS WITH ALGEBRA I
Thursday, 10/18/12
Today:
Return, go over Exam #1
Catch up on earlier slides:
Tangent and velocity problems
Introduction to limits
More on Limits
HW #3 will be posted later today or tomorrow
Reading:
Keep reading 2.2
Exercises (not to hand in):
2.2: 1-9 odd
Thursday, 10/18/12, Slide #1
Example: One-sided limits
 x 2 + 1, if x ≥ 0
f ( x) = 
− x − 2, if x < 0
Consider this function:
What is f(0)?
Look at the graph (on board)
What is limxÆ0 f(x) ?
What if we ask: As x gets close to, but x is
less than 0, what does f(x) get close to?
Or : As x gets close to, but x is greater than
0, what does f(x) get close to?
These are called left-handed and right-handed
limits as x approaches 0.
Thursday, 10/18/12, Slide #2
Definition of one-sided limits;
theorem about limits
“The limit of f(x) as x approaches a from the left is L”
means: As x approaches a (i.e.,
lim− f ( x) = L
x→a
gets closer and closer to a), but
x is less than a, f(x) approaches L.
“The limit of f(x) as x approaches a from the right is L”
means: As x approaches a,
but x is greater than a,
f(x) approaches L.
Theorem:
lim+ f ( x) = L
x→a
lim f ( x) = L if and only if lim− f ( x) = L and lim+ f ( x) = L
x→a
x→a
x→a
Thursday, 10/18/12, Slide #3
Exercise
For the piecewise-defined function given
below, what are: lim f ( x) and lim f ( x)
−
+
x →1
x →1
(State left limit first, then right)
A. 0 and 0
2

4
x
, if x < 1
B. 0 and 3

f ( x) = 0, if x = 1
C. 0 and 7
2 x + 5, if x > 1

D. 3 and 7
E. 7 and 3
Thursday, 10/18/12, Slide #4
Examples: Limits at Infinity
Whenever we use the symbol • or +•, we mean a value
that is becoming arbitrarily large and positive
Whenever we use the symbol -•, we mean a value that is
becoming arbitrarily large and negative
In graph below, f(-4) and f(4) are undefined. What are:
#1) limxÆ-4¯¯ f(x)
#2) limxÆ-4 f(x)
#3) limxÆ-4 f(x)
#4) limxÆ4¯¯ f(x)
#5) limxÆ4 f(x)
#6) limxÆ4 f(x)
#7) limxÆ-• f(x)
#8) limxÆ +• f(x)
Thursday, 10/18/12, Slide #5
Definition of limits using infinity
lim f ( x) = ∞
“The limit of f(x) as x approaches a is ¶”
means: As x approaches a, f(x) becomes
arbitrarily large and positive.
lim f ( x) = −∞
“The limit of f(x) as x approaches a is -¶”
means: As x approaches a, f(x) becomes
arbitrarily large and negative.
x→a
x→a
lim f ( x) = L
“The limit of f(x) as x approaches ¶ is L”
means: As x becomes arbitrarily large and
positive, f(x) approaches the number L.
Here are more limits using infinity:
x →∞
lim f ( x) = L,
x → −∞
Analogous definitions apply when x approaches a
from left and when x approaches a from the right
lim f ( x) = ∞, lim f ( x ) = −∞,
x →∞
x →∞
lim f ( x) = ∞, lim f ( x) = −∞
x → −∞
x → −∞
Thursday, 10/18/12, Slide #6