Estimating limits from graphs or tables - AP Calculus AB
Card 1 of 44
Consider the function 
. Which Reimann sum calculation would give the best approximation of the integral from 
to 
?
Consider the function
Tap to reveal answer
The mid-point Reimann sum is given by this formula:
, where 
, and 
is the number of
intervals. Thus, if the region from 
to 
is divided into twenty intervals, 
and 
. For ten intervals, 
and 
. For five internals, 
and 
. The higher the number of intervals, the more precise the estimation. Thus, when 
(and hence 
), the estimation is the most accurate.
The mid-point Reimann sum is given by this formula:
intervals. Thus, if the region from
← Didn't Know|Knew It →
What is the 
?
What is the
Tap to reveal answer
For this value, the Limit Laws can be applied:
For this value, the Limit Laws can be applied:
← Didn't Know|Knew It →
True or false: In the above graph of 
, the value of 
is 3.
True or false: In the above graph of
Tap to reveal answer
Removable discontinuities don't affect the limiting process. The limit process is essentially saying "As you get arbitrarily close to 
, the function is getting arbitrarily close to 
." As you can see, whether or not the function is defined at 
is irrelevant, because we're want to look at values close to it, but never exactly at 
.
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 
, you will end up at 
.
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.
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.
← Didn't Know|Knew It →
In the above graph of 
, evaluate the following limits:
In the above graph of
Tap to reveal answer
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 
from the right, what is 
getting close to?"
The same holds true for left limits, but you approach them from the other side.
In this problem, 
is a right limit. Following the above method, we find that as we approach 
from the right, we end up at 
. Similarly, 
is a left limit. Approaching 
from the left, we see that we arrive at 
.
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
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
← Didn't Know|Knew It →
Consider the function . Which Reimann sum calculation would give the best approximation of the integral from to ?
Consider the function
Tap to reveal answer
The mid-point Reimann sum is given by this formula:
, where , and is the number of
intervals. Thus, if the region from to is divided into twenty intervals, and . For ten intervals, and . For five internals, and . The higher the number of intervals, the more precise the estimation. Thus, when (and hence ), the estimation is the most accurate.
The mid-point Reimann sum is given by this formula:
intervals. Thus, if the region from
← Didn't Know|Knew It →
What is the ?
What is the
Tap to reveal answer
For this value, the Limit Laws can be applied:
For this value, the Limit Laws can be applied:
← Didn't Know|Knew It →
True or false: In the above graph of , the value of is 3.
True or false: In the above graph of
Tap to reveal answer
Removable discontinuities don't affect the limiting process. The limit process is essentially saying "As you get arbitrarily close to , the function is getting arbitrarily close to ." As you can see, whether or not the function is defined at is irrelevant, because we're want to look at values close to it, but never exactly at .
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 , you will end up at .
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.
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.
← Didn't Know|Knew It →
In the above graph of , evaluate the following limits:
In the above graph of
Tap to reveal answer
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 from the right, what is getting close to?"
The same holds true for left limits, but you approach them from the other side.
In this problem, is a right limit. Following the above method, we find that as we approach from the right, we end up at . Similarly, is a left limit. Approaching from the left, we see that we arrive at .
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
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
← Didn't Know|Knew It →
Consider the function . Which Reimann sum calculation would give the best approximation of the integral from to ?
Consider the function
Tap to reveal answer
The mid-point Reimann sum is given by this formula:
, where , and is the number of
intervals. Thus, if the region from to is divided into twenty intervals, and . For ten intervals, and . For five internals, and . The higher the number of intervals, the more precise the estimation. Thus, when (and hence ), the estimation is the most accurate.
The mid-point Reimann sum is given by this formula:
intervals. Thus, if the region from
← Didn't Know|Knew It →
What is the ?
What is the
Tap to reveal answer
For this value, the Limit Laws can be applied:
For this value, the Limit Laws can be applied:
← Didn't Know|Knew It →
True or false: In the above graph of , the value of is 3.
True or false: In the above graph of
Tap to reveal answer
Removable discontinuities don't affect the limiting process. The limit process is essentially saying "As you get arbitrarily close to , the function is getting arbitrarily close to ." As you can see, whether or not the function is defined at is irrelevant, because we're want to look at values close to it, but never exactly at .
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 , you will end up at .
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.
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.
← Didn't Know|Knew It →
In the above graph of , evaluate the following limits:
In the above graph of
Tap to reveal answer
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 from the right, what is getting close to?"
The same holds true for left limits, but you approach them from the other side.
In this problem, is a right limit. Following the above method, we find that as we approach from the right, we end up at . Similarly, is a left limit. Approaching from the left, we see that we arrive at .
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
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
← Didn't Know|Knew It →
Consider the function . Which Reimann sum calculation would give the best approximation of the integral from to ?
Consider the function
Tap to reveal answer
The mid-point Reimann sum is given by this formula:
, where , and is the number of
intervals. Thus, if the region from to is divided into twenty intervals, and . For ten intervals, and . For five internals, and . The higher the number of intervals, the more precise the estimation. Thus, when (and hence ), the estimation is the most accurate.
The mid-point Reimann sum is given by this formula:
intervals. Thus, if the region from
← Didn't Know|Knew It →
What is the ?
What is the
Tap to reveal answer
For this value, the Limit Laws can be applied:
For this value, the Limit Laws can be applied:
← Didn't Know|Knew It →
True or false: In the above graph of , the value of is 3.
True or false: In the above graph of
Tap to reveal answer
Removable discontinuities don't affect the limiting process. The limit process is essentially saying "As you get arbitrarily close to , the function is getting arbitrarily close to ." As you can see, whether or not the function is defined at is irrelevant, because we're want to look at values close to it, but never exactly at .
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 , you will end up at .
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.
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.
← Didn't Know|Knew It →
In the above graph of , evaluate the following limits:
In the above graph of
Tap to reveal answer
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 from the right, what is getting close to?"
The same holds true for left limits, but you approach them from the other side.
In this problem, is a right limit. Following the above method, we find that as we approach from the right, we end up at . Similarly, is a left limit. Approaching from the left, we see that we arrive at .
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
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
← Didn't Know|Knew It →
What is the ?
What is the
Tap to reveal answer
For this value, the Limit Laws can be applied:
For this value, the Limit Laws can be applied:
← Didn't Know|Knew It →
Consider the function . Which Reimann sum calculation would give the best approximation of the integral from to ?
Consider the function
Tap to reveal answer
The mid-point Reimann sum is given by this formula:
, where , and is the number of
intervals. Thus, if the region from to is divided into twenty intervals, and . For ten intervals, and . For five internals, and . The higher the number of intervals, the more precise the estimation. Thus, when (and hence ), the estimation is the most accurate.
The mid-point Reimann sum is given by this formula:
intervals. Thus, if the region from
← Didn't Know|Knew It →
True or false: In the above graph of , the value of is 3.
True or false: In the above graph of
Tap to reveal answer
Removable discontinuities don't affect the limiting process. The limit process is essentially saying "As you get arbitrarily close to , the function is getting arbitrarily close to ." As you can see, whether or not the function is defined at is irrelevant, because we're want to look at values close to it, but never exactly at .
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 , you will end up at .
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.
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.
← Didn't Know|Knew It →
In the above graph of , evaluate the following limits:
In the above graph of
Tap to reveal answer
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 from the right, what is getting close to?"
The same holds true for left limits, but you approach them from the other side.
In this problem, is a right limit. Following the above method, we find that as we approach from the right, we end up at . Similarly, is a left limit. Approaching from the left, we see that we arrive at .
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
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
← Didn't Know|Knew It →