Continuity in Terms of Limits - AP Calculus BC
Card 0 of 6
The graph above is a sketch of the function 
. For what intervals is 
continuous?
The graph above is a sketch of the function
For a function to be continuous at a point 
, 
must exist and 
.
This is true for all values of 
except 
and 
.
Therefore, the interval of continuity is 
.
For a function to be continuous at a point
This is true for all values of
Therefore, the interval of continuity is
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Consider the piecewise function:
What is 
?
Consider the piecewise function:
What is
The piecewise function
indicates that 
is one when 
is less than five, and is zero if the variable is greater than five. At 
, there is a hole at the end of the split.
The limit does not indicate whether we want to find the limit from the left or right, which means that it is necessary to check the limit from the left and right. From the left to right, the limit approaches 1 as 
approaches negative five. From the right, the limit approaches zero as 
approaches negative five.
Since the limits do not coincide, the limit does not exist for 
.
The piecewise function
indicates that
The limit does not indicate whether we want to find the limit from the left or right, which means that it is necessary to check the limit from the left and right. From the left to right, the limit approaches 1 as
Since the limits do not coincide, the limit does not exist for
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Consider the function 
.
Which of the following statements are true about this function?
I.
II. 
III. 
Consider the function
Which of the following statements are true about this function?
I.
II.
III.
For a function to be continuous at a particular point, the limit of the function at that point must be equal to the value of the function at that point.
First, notice that
.
This means that the function is continuous everywhere.
Next, we must compute the limit. Factor and simplify f(x) to help with the calculation of the limit.
Thus, the limit as x approaches three exists and is equal to 
, so I and II are true statements.
For a function to be continuous at a particular point, the limit of the function at that point must be equal to the value of the function at that point.
First, notice that
This means that the function is continuous everywhere.
Next, we must compute the limit. Factor and simplify f(x) to help with the calculation of the limit.
Thus, the limit as x approaches three exists and is equal to
Compare your answer with the correct one above
The graph above is a sketch of the function . For what intervals is continuous?
The graph above is a sketch of the function
For a function to be continuous at a point , must exist and .
This is true for all values of except and .
Therefore, the interval of continuity is .
For a function to be continuous at a point
This is true for all values of
Therefore, the interval of continuity is
Compare your answer with the correct one above
Consider the piecewise function:
What is ?
Consider the piecewise function:
What is
The piecewise function
indicates that is one when is less than five, and is zero if the variable is greater than five. At , there is a hole at the end of the split.
The limit does not indicate whether we want to find the limit from the left or right, which means that it is necessary to check the limit from the left and right. From the left to right, the limit approaches 1 as approaches negative five. From the right, the limit approaches zero as approaches negative five.
Since the limits do not coincide, the limit does not exist for .
The piecewise function
indicates that
The limit does not indicate whether we want to find the limit from the left or right, which means that it is necessary to check the limit from the left and right. From the left to right, the limit approaches 1 as
Since the limits do not coincide, the limit does not exist for
Compare your answer with the correct one above
Consider the function .
Which of the following statements are true about this function?
I.
II.
III.
Consider the function
Which of the following statements are true about this function?
I.
II.
III.
For a function to be continuous at a particular point, the limit of the function at that point must be equal to the value of the function at that point.
First, notice that
.
This means that the function is continuous everywhere.
Next, we must compute the limit. Factor and simplify f(x) to help with the calculation of the limit.
Thus, the limit as x approaches three exists and is equal to , so I and II are true statements.
For a function to be continuous at a particular point, the limit of the function at that point must be equal to the value of the function at that point.
First, notice that
This means that the function is continuous everywhere.
Next, we must compute the limit. Factor and simplify f(x) to help with the calculation of the limit.
Thus, the limit as x approaches three exists and is equal to
Compare your answer with the correct one above