All AP Calculus BC Resources
Example Questions
Example Question #1 : Limits
Evaluate the following limit:
The limit does not exist
The limit we are given is one sided, meaning we are approaching our x value from one side; in this case, the negative sign exponent indicates that we are approaching 3 from the left side, or using values slightly less than three on approach.
This corresponds to the part of the piecewise function for values less than 3. When we substitute our x value being approached, we get
Example Question #1 : Estimating Limits From Graphs And Tables
For the piecewise function:
, find .
Any real number.
Does not exist.
The limit indicates that we are trying to find the value of the limit as approaches to zero from the right side of the graph.
From right to left approaching , the limit approaches to 1 even though the value at of the piecewise function does not exist.
The answer is .
Example Question #1 : Estimating Limits From Graphs And Tables
Given the graph of above, what is ?
Examining the graph of the function above, we need to look at three things:
1) What is the limit of the function as it approaches zero from the left?
2) What is the limit of the function as it approaches zero from the right?
3) What is the function value at zero and is it equal to the first two statements?
If we look at the graph we see that as approaches zero from the left the values approach zero as well. This is also true if we look the values as approaches zero from the right. Lastly we look at the function value at zero which in this case is also zero.
Therefore, we can observe that as approaches .
Example Question #1 : Estimating Limits From Graphs And Tables
Given the graph of above, what is ?
Does not exist
Does not exist
Examining the graph above, we need to look at three things:
1) What is the limit of the function as approaches zero from the left?
2) What is the limit of the function as approaches zero from the right?
3) What is the function value as and is it the same as the result from statement one and two?
Therefore, we can determine that does not exist, since approaches two different limits from either side : from the left and from the right.
Example Question #1 : Estimating Limits From Graphs And Tables
Given the above graph of , what is ?
Examining the graph, we want to find where the graph tends to as it approaches zero from the right hand side. We can see that there appears to be a vertical asymptote at zero. As the x values approach zero from the right the function values of the graph tend towards positive infinity.
Therefore, we can observe that as approaches from the right.
Example Question #1 : Limits
Given the above graph of , what is ?
Does Not Exist
Does Not Exist
Examining the graph, we can observe that does not exist, as is not continuous at . We can see this by checking the three conditions for which a function is continuous at a point :

A value exists in the domain of

The limit of exists as approaches

The limit of at is equal to
Given , we can see that condition #1 is not satisfied because the graph has a vertical asymptote instead of only one value for and is therefore an infinite discontinuity at .
We can also see that condition #2 is not satisfied because approaches two different limits: from the left and from the right.
Based on the above, condition #3 is also not satisfied because is not equal to the multiple values of .
Thus, does not exist.