AP Calculus AB : Estimating limits from graphs or tables
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Example Question #1 : Estimating Limits From Graphs Or Tables
Consider the function 
The mid-point Reimann sum is given by this formula:
intervals. Thus, if the region from 
Example Question #1 : Estimating Limits From Graphs Or Tables
What is the 
Does not exist
For this value, the Limit Laws can be applied:
Example Question #3 : Estimating Limits From Graphs Or Tables
True or false: In the above graph of 
True: The removable discontinuity does not affect the limit, and the right and left limits evaluate to 3.
False: We can't take the limit where the function isn't defined.
False: The left and right limits exist, but neither of them is three.
False: The left and right limits exist, but exactly one of them isn't three.
False: Both the left and right limits are three, but the regular limit isn't.
True: The removable discontinuity does not affect the limit, and the right and left limits evaluate to 3.
Removable discontinuities don't affect the limiting process. The limit process is essentially saying "As you get arbitrarily close to 
Graphically, we can see that the left and right limits are both three. If you trace the graph from either the left or right of 
By definition, if both the right and left limit evaluate to the same thing, the actual limit must agree. It's not possible for both of them to be three, but for the limit to disagree with them (at least in a two dimensional graph!)
Thus the correct answer is that the limit exists, and is three.
Example Question #4 : Estimating Limits From Graphs Or Tables
In the above graph of 
A right limit is found graphically by starting at a point to the right of the specified x value and tracing along the graph to the left until you hit the specified x. Regardless of whether or not the function is defined, or perhaps if its defined, but not where your finger ends up, your finger should be at the limit. This is because limits don't concern themselves with what happens at the specified x value. For right limits, we're essentially asking "If I get arbitrarily close to 
The same holds true for left limits, but you approach them from the other side.
In this problem, 
Lastly, a regular limit is defined when both the right and left limits are defined and equal. In this case, the right limit is 5 and the left limit is 3, so the regular limit does not exist
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