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592 PowerPoint Assignment

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592 PowerPoint Assignment
Limits of Functions
What do we mean by limit?

In everyday conversation, we use the
word “limit” to describe the ultimate
behavior of something.
e.g. “the limit of one’s patience”
Limit of a Function
lim f ( x )
x c
Is the number that f(x) approaches as x
approaches c.
So…we are only concerned about what happens
with f(x) as x gets arbitrarily close to c.
We don’t actually care about what happens when
x=c.
How do we find limits?


There are several methods, but we
will use the TABLE METHOD.
So let’s use this method…
Our First Example
lim ( 2 x  4)
x 3
The left column of our table will contain numbers
approaching 3.
The right column of our table will contain the value of
these numbers evaluated in the function 2x+4
How do we approach 3?

We have two choices:
1) Approach 3 from the left
2) Approach 3 from the right


Approaching from the left will give the
Left-Sided Limit
Approaching from the right will give the
Right-Sided Limit
The Left-Sided Limit
x
2x+4
2.9
9.8
2.99
9.98
2.999
9.998
2.9999
9.9998
2.99999
9.99998
The column on the left
contains the values we chose
that are LESS than 3 and
are approaching 3.
The values in the right
column are approaching 10.
Thus, 10 is the Left
Sided Limit!
The Right-Sided Limit
x
2x+4
3.1
10.2
3.01
10.02
3.001
10.002
3.0001
10.0002
3.00001
10.00002
The column on the left
contains the values we chose
that are GREATER than 3
and are approaching 3.
The values in the right
column are approaching 10.
Thus, 10 is the RightSided Limit!
Are we done?


Almost… We were simply asked to
find the limit of the function, not the
left or right sided limits.
However, since the left and right
sided limits both were equal to 3, we
say that the limit of the function is 3.
One more thing…


If the Left-Sided Limit and RightSided Limit of a function do not
agree, then we say that the limit of
the function DOES NOT EXIST.
In conclusion, remember to always…
Fly UP