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OBJECTIVES:
1. Estimate limits of functions at a point.
2. Estimate limits of functions at infinity.
2.
Estimate limits of functions at a point.
Estimate limits of functions at infinity.
How can I find the limits of a function numerically
and graphically?
2
3.
State the end behavior
by filling in the blanks.
4.
State the end behavior
by filling in the blanks.
5.
Informal Definition:
If f(x) becomes arbitrarily
close to a single REAL
number L as x approaches
c from either side, the
limit of f(x), as x
approaches c, is L.
5
6.
The limit of f(x)…
is L.
Notation: lim f x L
x c
as x approaches c…
x
c
7.
There are three approaches to finding a limit:
1. Numerical Approach – Construct a table of
values
This
Lesson
2. Graphical Approach – Draw a graph
3. Analytic Approach – Use Algebra or
8.
Complete the table to find the limit (if it exists).
3
lim x
x 2
x 1.9 1.99 1.999 2 2.001 2.01 2.1
f(x) 6.859 7.88 7.988 8 8.012 8.12 9.261
If the function is continuous at the value of x,
the limit is easy to calculate.
3
lim x 8
x 2
10.
Complete the table to find the limit (if it exists).
x2 1
lim x 1 Can’t divide by 0
x 1
x -1.1 -1.01 -1.001 -1 -.999 -.99 -.9
f(x) -2.1 -2.01 -2.001 DNE -1.999 -1.99 -1.9
If the function is not continuous at the value of
x, a graph and table can be very useful.
x2 1
lim x 1 2
x 1
12.
f(x) approaches a different number from the
right side of c than it approaches from the
left side.
lim f x Does Not Exist
x 4
13.
f(x) increases or decreases without bound as
x approaches c.
lim f x Does Not Exist
x 0
14.
f(x) oscillates between two fixed values as x
approaches c.
Closest
Closer
Close
x 2 2 2 2 2 2
3 5 0 5 3
f(x) DNE
-1 1 -1 1 -1 1
lim sin 1x Does Not Exist
x 0
15.
If the domain is restricted (not infinite), the
limit of f(x) exists as x approaches an
endpoint of the domain.
lim f x 5
x 5
17.
Given the function t defined by the graph, find the limits at right.
t x 1. lim t x 2
x 4
2. lim t x 3
x 3
3. lim t x DNE
x 0
4. lim t x
x 6
3
5. lim t x DNE
x 2
6. lim t x 2
x 5
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