6. Intuitive Limits and Continuity

c. Tutorial on Limits and Continuity

1. The limit from the left at .
   

   Use the curve on the left of . The limit from the left is the height \(f(x)\) approaches as \(x\) approaches from the left, i.e. \(x\) is less than but getting very close to .

   Perfect! because the curve on the left goes to height as \(x\) goes to .

   Sorry. Read the hint. Try again.

   I can't understand your answer. It may have been typed wrong. I am expecting an integer, infinity, -infinity or undefined.

   You need to enter an answer before checking.

   because the curve on the left goes to height as \(x\) goes to .

2. The limit from the right at .
   

   Use the curve on the right of . The limit from the right is the height \(f(x)\) approaches as \(x\) approaches from the right, i.e. \(x\) is greater than but getting very close to .

   Excellent! because the curve on the right goes to height as \(x\) goes to .

   Sorry. Read the hint. Try again.

   I can't understand your answer. It may have been typed wrong. I am expecting an integer, infinity, -infinity or undefined.

   You need to enter an answer before checking.

   because the curve on the right goes to height as \(x\) goes to .

3. The function value at . Enter a number or undefined.
   

   is the value of \(f(x)\) at if it is defined. This is the solid dot above if there is one.

   Good Job! because the solid dot above is at height .

   Way to go! is undefined because there is no solid dot above .

   Sorry. Read the hint. Try again.

   I can't understand your answer. It may have been typed wrong. I am expecting an integer, infinity, -infinity or undefined.

   You need to enter an answer before checking.

   because the solid dot above is at height .

4. Does the (\(2\)-sided) limit, , exist?
   Yes No

   The (2-sided) limit exists when the limit from the left is equal to the limit from the right.

   That's right! exists because .

   Perfect! but . They are not equal.

   Sorry. Read the hint. Try again.

   I can't understand your answer. It may have been typed wrong. I am expecting an integer, infinity, -infinity or undefined.

   You need to enter an answer before checking.

   exists because .

   does not exist because while which are not equal.

5. If the (\(2\)-sided) limit exists, give its value. If it does not exist, enter infinity, -infinity or undefined.
   \(=\,\)

   When the (2-sided) limit exists, it is equal to the limit from the left and the limit from the right.
   When it does not exist, it is \(\infty\) when the graph goes up forever on both sides and it is \(-\infty\) when the graph goes down forever on both sides. Otherwise, it is undefined.

   That's right! \(=\;\) because .

   Way to go! does not exist and is neither \(\infty\) nor \(-\infty\).

   Sorry. Read the hint. Try again.

   I can't understand your answer. It may have been typed wrong. I am expecting an integer, infinity, -infinity or undefined.

   You need to enter an answer before checking.

   \(=\) because .

   is undefined because it does not exist and it is neither \(\infty\) nor \(-\infty\).

6. Is \(f(x)\) continuous from the left at ?
   Yes No

   \(f(x)\) is continuous from the left when the limit from the left is equal to the value of the function.

   Super! \(f(x)\) is continuous from the left because .

   On the mark! \(f(x)\) is not continuous from the left because but . They are not equal.

   Sorry. Read the hint. Try again.

   I can't understand your answer. It may have been typed wrong. I am expecting an integer, infinity, -infinity or undefined.

   You need to enter an answer before checking.

   \(f(x)\) is continuous from the left because .

   \(f(x)\) is not continuous from the left because while which are not equal.

7. Is \(f(x)\) continuous from the right at ?
   Yes No

   \(f(x)\) is continuous from the right when the limit from the right is equal to the value of the function.

   Super! \(f(x)\) is continuous from the right because .

   On the mark! \(f(x)\) is not continuous from the right because but . They are not equal.

   Sorry. Read the hint. Try again.

   I can't understand your answer. It may have been typed wrong. I am expecting an integer, infinity, -infinity or undefined.

   You need to enter an answer before checking.

   \(f(x)\) is continuous from the right because .

   \(f(x)\) is not continuous from the right because while which are not equal.

8. Is \(f(x)\) continuous at ?
   Yes No

   \(f(x)\) is continuous when the limit from the left, the limit from the right and the value of the function are all equal.

   Super! \(f(x)\) is continuous from the right because .

   On the mark! \(f(x)\) is not continuous from the right because but . They are not equal.

   Sorry. Read the hint. Try again.

   I can't understand your answer. It may have been typed wrong. I am expecting an integer, infinity, -infinity or undefined.

   You need to enter an answer before checking.

   \(f(x)\) is continuous because .

   \(f(x)\) is not continuous because and and which are not all equal.

1. The limit from the left at .
   

   The limit from the left is the value \(f(x)\) approaches if \(x \lt\) but getting closer to . So we successively plug , into the formula for \(x \lt\) . The limit is what we get by plugging \(x=\) into the formula for \(x \lt\) .

   Perfect! because this is the result when one plugs into the formula for \(x \lt\) .

   Sorry. Read the hint. Try again.

   I can't understand your answer. It may have been typed wrong. I am expecting an integer, infinity, -infinity or undefined.

   You need to enter an answer before checking.

   because this is the result when one plugs into the formula for \(x \lt\) .

2. The limit from the right at .
   .

   The limit from the right is the value \(f(x)\) approaches if \(x \gt\) but getting closer to . So we successively plug , into the formula for \(x \gt\) . The limit is what we get by plugging into the formula for \(x \gt\) .

   Impressive! because this is the result when one plugs into the formula for \(x \gt\) .

   Sorry. Read the hint. Try again.

   I can't understand your answer. It may have been typed wrong. I am expecting an integer, infinity, -infinity or undefined.

   You need to enter an answer before checking.

   because this is the result when one plugs into the formula for \(x \gt\) .

3. The function value at . Enter a number or undefined.
   

   The value of \(f(x)\) at is if it is defined. This is the value next to if there is one.

   Good Job! because this is the value next to in the formula.

   Way to go! is undefined because there is no value for .

   Sorry. Read the hint. Try again.

   I can't understand your answer. It may have been typed wrong. I am expecting an integer, infinity, -infinity or undefined.

   You need to enter an answer before checking.

   because this is the value next to in the formula.

4. Does the (\(2\)-sided) limit, , exist?
   Yes No

   The (2-sided) limit exists when the limit from the left is equal to the limit from the right.

   That's right! exists because .

   Perfect! but . They are not equal.

   Sorry. Read the hint. Try again.

   I can't understand your answer. It may have been typed wrong. I am expecting an integer, infinity, -infinity or undefined.

   You need to enter an answer before checking.

   exists because .

   does not exist because while which are not equal.

5. If the (\(2\)-sided) limit exists, give its value. If it does not exist, enter infinity, -infinity or undefined.
   

   When the (2-sided) limit exists, it is equal to the limit from the left and the limit from the right.
   When it does not exist, it is \(\infty\) when the function approaches \(\infty\) on both sides and it is \(-\infty\) when the function approaches \(-\infty\) on both sides. Otherwise, it is undefined.

   That's right! \(=\) because .

   Way to go! does not exist and is not \(\infty\) nor \(-\infty\).

   Sorry. Read the hint. Try again.

   I can't understand your answer. It may have been typed wrong. I am expecting an integer, infinity, -infinity or undefined.

   You need to enter an answer before checking.

   \(=\) because .

   is undefined because it does not exist and it is neither \(\infty\) nor \(-\infty\).

6. Is \(f(x)\) continuous from the left at ?
   Yes No

   \(f(x)\) is continuous from the left when the limit from the left is equal to the value of the function.

   Super! \(f(x)\) is continuous from the left because .

   On the mark! \(f(x)\) is not continuous from the left because but . They are not equal.

   Sorry. Read the hint. Try again.

   I can't understand your answer. It may have been typed wrong. I am expecting an integer, infinity, -infinity or undefined.

   You need to enter an answer before checking.

   \(f(x)\) is continuous from the left because .

   \(f(x)\) is not continuous from the left because while which are not equal.

7. Is \(f(x)\) continuous from the right at ?
   Yes No

   \(f(x)\) is continuous from the right when the limit from the right is equal to the value of the function.

   Super! \(f(x)\) is continuous from the right because .

   On the mark! \(f(x)\) is not continuous from the right because but . They are not equal.

   Sorry. Read the hint. Try again.

   I can't understand your answer. It may have been typed wrong. I am expecting an integer, infinity, -infinity or undefined.

   You need to enter an answer before checking.

   \(f(x)\) is continuous from the right because .

   \(f(x)\) is not continuous from the right because while which are not equal.

8. Is \(f(x)\) continuous at ?
   Yes No

   \(f(x)\) is continuous when the limit from the left, the limit from the right and the value of the function are all equal.

   Super! \(f(x)\) is continuous because .

   On the mark! \(f(x)\) is not continuous because and and . They are not all equal.

   Sorry. Read the hint. Try again.

   I can't understand your answer. It may have been typed wrong. I am expecting an integer, infinity, -infinity or undefined.

   You need to enter an answer before checking.

   \(f(x)\) is continuous because .

   \(f(x)\) is not continuous because and and which are not all equal.