Limits Overview
Limit Notation (Left-Hand, Right-Hand, Overall)
Finding Limits Using a Graph
Finding a Limit Using Equations
Continuity

Example 1: Continuity Using a Graph

Given the graph of f ( x ) below, determine the intervals where the function is continuous.

Any place where it is not continuous state why.

 

Step 1: Identify all the x-values where you would have to lift your pencil in order to draw the given graph.

 

Starting on the left side of the graph and drawing to the right, you can see that you would need to lift your pencil at x = -1 where there is a jump in the graph, and x = 2 when you would have to lift your pencil up to put the dot on the graph at (2, 2), and again to get it back on the graph.

 

This tells us that our graph is NOT continuous at x = -1 and x = 2.

 

Step 2 (Optional): Show why those x-values are not continuous.

 

Determine which part of the definition failed:

 

lim x c   f ( x ) = f ( c )

 

For x = -1 we can see that the overall limit does not exist because the left-hand limit does not equal the right-hand limit. We also have no actual function value at x = -1, f ( 1 ) = Undefined .

 

For x = 2 we can see that the overall limit does exist because the left-hand limit does equal the right-hand limit.

lim x 2   f ( x ) = 4

We also have a function value at x = 2, f ( 2 ) = 2 .

However, those two values do not equal each other.

x = 1

Left-Hand Limit Right-Hand Limit

lim x 1   f ( x ) = DNE

 

x = 2

Left-Hand Limit = Right-Hand Limit

lim x 2   f ( x ) = 4

f ( 2 ) = 2

lim x 2   f ( x ) = 4 f ( 2 ) = 2

lim x 2   f ( x ) f ( 2 )

Final Result:

The function, f ( x ) , is continuous on the x-interval : ( , 1) ( 1 , 2) (2 , )

It is not continuous at x = -1 because there is a jump in the graph.

It is not continuous at x = 2 because there is a hole in the graph.

 

Notes:

          Notice you will use the x-values where the function is not continuous as the places the intervals break apart. Even though you are being asked to find where the function is continuous. The answer will always be found by determining where it is not continuous.

          You will usually want to provide your solution in set notation. In set notation (parenthesis) mean that value is (not included), and [brackets] mean that the value [is included].

          In set notation you connect your intervals with a union symbol, .

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