Limits of Functions (including one-sided limits) - AP Calculus AB
Card 1 of 770
Which of the following is equal to 
?
Which of the following is equal to
Tap to reveal answer
The limit of a function as 
approaches a value 
exists if and only if the limit from the left is equal to the limit from the right; the actual value of 
is irrelevant. Since the function is piecewise-defined, we can determine whether these limits are equal by finding the limits of the individual expressions. These are both polynomials, so the limits can be calculated using straightforward substitution:
, so 
.
The limit of a function as
← Didn't Know|Knew It →
Evaluate the limit:
Evaluate the limit:
Tap to reveal answer
To evaluate the limit, we must first determine whether the limit is right or left sided; the minus sign exponent indicates we are approaching three from the left, or using values slightly less than three. These correspond to the first part of the piecewise function, whose limit is 9 (substitution).
To evaluate the limit, we must first determine whether the limit is right or left sided; the minus sign exponent indicates we are approaching three from the left, or using values slightly less than three. These correspond to the first part of the piecewise function, whose limit is 9 (substitution).
← Didn't Know|Knew It →
Which of the following is equal to ?
Which of the following is equal to
Tap to reveal answer
The limit of a function as approaches a value exists if and only if the limit from the left is equal to the limit from the right; the actual value of is irrelevant. Since the function is piecewise-defined, we can determine whether these limits are equal by finding the limits of the individual expressions. These are both polynomials, so the limits can be calculated using straightforward substitution:
, so .
The limit of a function as
← Didn't Know|Knew It →
Evaluate the limit:
Evaluate the limit:
Tap to reveal answer
To evaluate the limit, we must first determine whether the limit is right or left sided; the minus sign exponent indicates we are approaching three from the left, or using values slightly less than three. These correspond to the first part of the piecewise function, whose limit is 9 (substitution).
To evaluate the limit, we must first determine whether the limit is right or left sided; the minus sign exponent indicates we are approaching three from the left, or using values slightly less than three. These correspond to the first part of the piecewise function, whose limit is 9 (substitution).
← Didn't Know|Knew It →
Which of the following is equal to ?
Which of the following is equal to
Tap to reveal answer
The limit of a function as approaches a value exists if and only if the limit from the left is equal to the limit from the right; the actual value of is irrelevant. Since the function is piecewise-defined, we can determine whether these limits are equal by finding the limits of the individual expressions. These are both polynomials, so the limits can be calculated using straightforward substitution:
, so .
The limit of a function as
← Didn't Know|Knew It →
Evaluate the limit:
Evaluate the limit:
Tap to reveal answer
To evaluate the limit, we must first determine whether the limit is right or left sided; the minus sign exponent indicates we are approaching three from the left, or using values slightly less than three. These correspond to the first part of the piecewise function, whose limit is 9 (substitution).
To evaluate the limit, we must first determine whether the limit is right or left sided; the minus sign exponent indicates we are approaching three from the left, or using values slightly less than three. These correspond to the first part of the piecewise function, whose limit is 9 (substitution).
← Didn't Know|Knew It →
Evaluate the limit:
Evaluate the limit:
Tap to reveal answer
To evaluate the limit, we must first determine whether the limit is right or left sided; the minus sign exponent indicates we are approaching three from the left, or using values slightly less than three. These correspond to the first part of the piecewise function, whose limit is 9 (substitution).
To evaluate the limit, we must first determine whether the limit is right or left sided; the minus sign exponent indicates we are approaching three from the left, or using values slightly less than three. These correspond to the first part of the piecewise function, whose limit is 9 (substitution).
← Didn't Know|Knew It →
Evaluate the limit:
Evaluate the limit:
Tap to reveal answer
To evaluate the limit, we must first determine whether the limit is right or left sided; the minus sign exponent indicates we are approaching three from the left, or using values slightly less than three. These correspond to the first part of the piecewise function, whose limit is 9 (substitution).
To evaluate the limit, we must first determine whether the limit is right or left sided; the minus sign exponent indicates we are approaching three from the left, or using values slightly less than three. These correspond to the first part of the piecewise function, whose limit is 9 (substitution).
← Didn't Know|Knew It →
Which of the following is equal to ?
Which of the following is equal to
Tap to reveal answer
The limit of a function as approaches a value exists if and only if the limit from the left is equal to the limit from the right; the actual value of is irrelevant. Since the function is piecewise-defined, we can determine whether these limits are equal by finding the limits of the individual expressions. These are both polynomials, so the limits can be calculated using straightforward substitution:
, so .
The limit of a function as
← Didn't Know|Knew It →
Evaluate the limit:
Evaluate the limit:
Tap to reveal answer
To evaluate the limit, we must first determine whether the limit is right or left sided; the minus sign exponent indicates we are approaching three from the left, or using values slightly less than three. These correspond to the first part of the piecewise function, whose limit is 9 (substitution).
To evaluate the limit, we must first determine whether the limit is right or left sided; the minus sign exponent indicates we are approaching three from the left, or using values slightly less than three. These correspond to the first part of the piecewise function, whose limit is 9 (substitution).
← Didn't Know|Knew It →
Which of the following is equal to ?
Which of the following is equal to
Tap to reveal answer
The limit of a function as approaches a value exists if and only if the limit from the left is equal to the limit from the right; the actual value of is irrelevant. Since the function is piecewise-defined, we can determine whether these limits are equal by finding the limits of the individual expressions. These are both polynomials, so the limits can be calculated using straightforward substitution:
, so .
The limit of a function as
← Didn't Know|Knew It →
Which of the following is equal to ?
Which of the following is equal to
Tap to reveal answer
The limit of a function as approaches a value exists if and only if the limit from the left is equal to the limit from the right; the actual value of is irrelevant. Since the function is piecewise-defined, we can determine whether these limits are equal by finding the limits of the individual expressions. These are both polynomials, so the limits can be calculated using straightforward substitution:
, so .
The limit of a function as
← Didn't Know|Knew It →
Which of the following is equal to ?
Which of the following is equal to
Tap to reveal answer
The limit of a function as approaches a value exists if and only if the limit from the left is equal to the limit from the right; the actual value of is irrelevant. Since the function is piecewise-defined, we can determine whether these limits are equal by finding the limits of the individual expressions. These are both polynomials, so the limits can be calculated using straightforward substitution:
, so .
The limit of a function as
← Didn't Know|Knew It →
Evaluate the limit:
Evaluate the limit:
Tap to reveal answer
To evaluate the limit, we must first determine whether the limit is right or left sided; the minus sign exponent indicates we are approaching three from the left, or using values slightly less than three. These correspond to the first part of the piecewise function, whose limit is 9 (substitution).
To evaluate the limit, we must first determine whether the limit is right or left sided; the minus sign exponent indicates we are approaching three from the left, or using values slightly less than three. These correspond to the first part of the piecewise function, whose limit is 9 (substitution).
← Didn't Know|Knew It →
Evaluate the limit:
Evaluate the limit:
Tap to reveal answer
To evaluate the limit, we must first determine whether the limit is right or left sided; the minus sign exponent indicates we are approaching three from the left, or using values slightly less than three. These correspond to the first part of the piecewise function, whose limit is 9 (substitution).
To evaluate the limit, we must first determine whether the limit is right or left sided; the minus sign exponent indicates we are approaching three from the left, or using values slightly less than three. These correspond to the first part of the piecewise function, whose limit is 9 (substitution).
← Didn't Know|Knew It →
Which of the following is equal to ?
Which of the following is equal to
Tap to reveal answer
The limit of a function as approaches a value exists if and only if the limit from the left is equal to the limit from the right; the actual value of is irrelevant. Since the function is piecewise-defined, we can determine whether these limits are equal by finding the limits of the individual expressions. These are both polynomials, so the limits can be calculated using straightforward substitution:
, so .
The limit of a function as
← Didn't Know|Knew It →
Which of the following is equal to ?
Which of the following is equal to
Tap to reveal answer
The limit of a function as approaches a value exists if and only if the limit from the left is equal to the limit from the right; the actual value of is irrelevant. Since the function is piecewise-defined, we can determine whether these limits are equal by finding the limits of the individual expressions. These are both polynomials, so the limits can be calculated using straightforward substitution:
, so .
The limit of a function as
← Didn't Know|Knew It →
Which of the following is equal to ?
Which of the following is equal to
Tap to reveal answer
The limit of a function as approaches a value exists if and only if the limit from the left is equal to the limit from the right; the actual value of is irrelevant. Since the function is piecewise-defined, we can determine whether these limits are equal by finding the limits of the individual expressions. These are both polynomials, so the limits can be calculated using straightforward substitution:
, so .
The limit of a function as
← Didn't Know|Knew It →
Evaluate the limit:
Evaluate the limit:
Tap to reveal answer
To evaluate the limit, we must first determine whether the limit is right or left sided; the minus sign exponent indicates we are approaching three from the left, or using values slightly less than three. These correspond to the first part of the piecewise function, whose limit is 9 (substitution).
To evaluate the limit, we must first determine whether the limit is right or left sided; the minus sign exponent indicates we are approaching three from the left, or using values slightly less than three. These correspond to the first part of the piecewise function, whose limit is 9 (substitution).
← Didn't Know|Knew It →
Evaluate the limit:
Evaluate the limit:
Tap to reveal answer
To evaluate the limit, we must first determine whether the limit is right or left sided; the minus sign exponent indicates we are approaching three from the left, or using values slightly less than three. These correspond to the first part of the piecewise function, whose limit is 9 (substitution).
To evaluate the limit, we must first determine whether the limit is right or left sided; the minus sign exponent indicates we are approaching three from the left, or using values slightly less than three. These correspond to the first part of the piecewise function, whose limit is 9 (substitution).
← Didn't Know|Knew It →