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Definition of a Limit
Definition:
The notation means that as x approaches (or gets very close to) x_{0 }, the limit of the function f ( x ) gets very close to the value L.
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Basically, a limit is the value a function approaches at a certain point. Limits can be found by:
 plugging the xvalue into the equation
 looking at a graph and estimating the yvalue for a function at that point
 plugging the equation into a calculator and using a table to see what value the function approaches from the left and right sides
Write using limit notation:
 Write the limit of as approaches from the left.
 Write the limit of as approaches .
 Write the limit of as approaches .
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Find the following limits at x = 0:
1.
x  0.2  0.1  0.01  0  0.01  0.1  0.2 

f ( x )  2.993347  2.998334  2.999983  Undefined  2.999983  2.998334 
2.993347 
2.
x  0.2  0.1  0.01  0  0.01  0.1  0.2 

f ( x )  0.993347  0.998334  0.999983  Undefined  1.000001  1.000012  1.000027 
3.
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Find the following limits:
OneSided Limits
If the value that the function approaches differs on the left and the right, you can use onesided limits to determine the value.
What is the limit of this function as x approaches 0 from the left? From the right?
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Limits from the left are written with a  after the number, from the right has a +.
Tip: The sign corresponds to the sides of the yaxis. The right side is positive, the left is negative.
Does the Limit Exist?
For a limit to exist, the limit from the right side must be equal to the limit from the left. If the righthand limit does not equal the limit from the left then the limit does not exist. For example, in the graph above, . Therefore does not exist.
To determine if the limit of a piecewise function (a function with two or more parts) exists, you must see if the righthand and lefthand limits are equal.


Remember that we are not concerned about finding the value of f ( x ) at x but rather near x . So, for x < 1 (limit from the left),
and for x > 1 (limit from the right),
Now since the limit exists and is the same on both sides, it follows that
Practice
Find the following limits:



and


and
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Find the following limits based on the equation:
Hint: Graph the equations or look at a table.
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Determine if the limits exist:
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