learning log #7. 2.2 the limit of a function uses of the word limit in everyday life (speed) roughly...
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Learning Log #7
2.2 The Limit of a Function
Uses of the word “Limit” in everyday life (speed)
Roughly speaking, the “Limit” is the y-value that the function is approaching as x approaches a certain number.
Enter the seven functions into your calculator so we can explore the concept of a limit intuitively.
From the home screen, find
1 1 1
1 1 1
(.9), (.99), (.999),
(1.1), (1.01), (1.001)
Y Y Y
Y Y Y
1Now in L enter .9,.99,.999,1.001,1.01,1.1
2 2 1 1Define L to be L ( )Y L
2 2 1Now define L ( )Y L
2The table should show that Y gets closer
to 3 as x approaches 1.
1The table shows Y gets closer to 5
as approaches 1.
Do the same for the next five functions.
Write the conclusion for each limit in your notes for each problem in
mathematical notation.
Example of a graph from the whiteboard
The conclusions after looking at the table.1 5
1 1
2 61 1
3 71 1
41
lim ( ) 5 lim ( )
lim ( ) 3 lim ( )
lim ( ) 4 lim ( )
lim ( )
x x
x x
x x
x
Y x Y x DNE
Y x Y x DNE or
Y x Y x DNE
Y x DNE
7 71 1
One-sided Limits
lim ( ) lim ( )x x
Y x Y x
NOTES:
: lim ( )
"The limit of f(x) as x approaches a,
equals L"
x aDef f x L
alternate def:
lim ( )
( )x a
f x L
is f x L as x a
We only care about the values of the function near a. We do not care if there is a function value at x=a or if that value is the same as our limit.
Illustration:
The Limit is the same for all three.
:
lim ( )x a
DEFINITION
f x L
if and only if
lim ( ) lim ( )x a x a
f x L f x
If the left and right sided limits are
the same, then the limit exists. If the limit
exists, then the left and right sided limits
are the
:
same.
Verbal
2
Infinite Limits
1:Example y
x
20
20
1lim ,
1lim
expressing a particular way that
the limit does not exist.
x
x
DNE howeverx
wecan sayx
The values of f(x) increase without bound
as x becomes closer to 0.
: The line x = a is called a vertical asymptote
of f(x) if at least 1 of the statements is true:
Def
lim ( ) lim ( )
lim ( ) lim ( )
lim ( ) lim ( )
x a x a
x a x a
x a x a
f x f x
f x f x
f x f x
Draw a mini-graph for each scenario.
2.2 IN-CLASS PROBLEMS