8/24/2009
1
Calculus I
Chapter 2
Limits and Continuity
3 Ways to Evaluate a Limit
o
Numerically
o
Graphically
o
Analytically
Example:
o
Consider the function
o
What is the domain?
o
What happens when
x
"gets close" to 1?
n
Numerically
n
Graphically
1
1
)
(
2


=
x
x
x
f
What does it mean for a limit to
exist? (Section 2.4)
o
For
to exist we must have:
for some real number L.
o
That is, for the limit to exist, the limit from the left
and right must both exist and be equal.
)
(
lim
x
f
c
x
→
L
x
f
x
f
c
x
c
x
=
=
+

→
→
)
(
lim
)
(
lim
3 Things That Can Cause A Limit To
Not Exist
o
Limits from the right and left differ.
o
Unbounded behavior.
o
Oscillating behavior.
n
CAUTION: Just because a function oscillates does
not necessarily mean that the limit does not exist.
Consider
x
x
x
f
sin
)
(
=
Evaluating Limits Analytically
Section 2.2
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Limit Laws
Let b and c be real numbers and
let n be a positive integer. Let f
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 Spring '11
 PamelaSatterfield
 Continuity, Limits, Limit, lim, Limit of a function

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