Survey of Calculus
Fall 2002
Smith
1
2.1
Limits and Continuity
Limits
There are basically three ways to evaluate limits:
(1)
_____________________
(2) _____________________
(3) _____________________
I.
Evaluating limits
graphically
.
A.
The limit of f(x) as x approaches a certain number, c, is the ____________________ that the function gets
“___________________”
(i.e., the yvalue “gets closer to”) as x gets “close to” c (from both sides).
Note:
The limit of function is NOT the same as the functional value!
Example:
Consider the graph of
3
3
2
)
(
2
−
−
−
=
x
x
x
x
f
.
(You will need to put the hole in the graph.
I can’t get my
computer to do it for us.
Do you remember where it belongs from College Algebra?)
x
y
4
3
2
1
0
1
2
3
4
3
2
1
0
1
2
3
4
5
Notice that as x gets “closer to” 3 from both sides, the yvalue gets “closer to”
___________.
This means that the limit of f(x) as x approaches 3 is ______________.
B. Notation:
is read as “the limit of f(x) as x approaches c equals L”.
In the example above,
L
x
f
c
x
=
→
)
(
lim
)
(
lim
3
x
f
x
→
= ____________.
C.
If the limit of a function exists, it must be a single real number.
Otherwise, we say that the limit
______________________________(DNE).
We may use the symbols _____ and ________ to indicate that the
values of a function become arbitrarily large or arbitrarily small.
Note the paragraph on p. 89.
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Survey of Calculus
Fall 2002
Smith
2
One –Sided Limits
A.
means the yvalue that function gets “close to” as x gets “close to” c from the ______________.
)
(
lim
x
f
c
x
−
→
B.
means the yvalue that the function gets “close to” as x gets “close to” c from the _____________.
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 Fall '05
 PamelaSatterfield
 Continuity, Derivative, Limits

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