Calculus

Step 1: Check for continuity of at .
For to be continuous at , we need .
We know that . Let's find the limit: Since , we have .
By the Squeeze Theorem, . Since , is continuous at .
Step 2: Check for differentiability of at .
The derivative at is given by: Since , we have .
By the Squeeze Theorem, . So, , which means is differentiable at .
Step 3: Find the derivative for .
Using the product rule and the chain rule: Step 4: Check for continuity of at .
For to be continuous at , we need . We know that . However, does not exist.
Therefore, does not exist, which means is not continuous at .

Step 1: Analyze statement (1).
Statement (1) is generally false as the left side is a constant and the right side is a function of .
Step 2: Analyze statement (2).
The given statement is .
Using the property , we have .
So the statement becomes , which is false.
Step 3: Analyze statement (3).
Statement (3) is false because since is a constant.
Step 4: Analyze statement (4).
Since (1) and (2) are false as written, (4) is false.
Re-evaluation of Option (2):
If option (2) was intended to be , then it would be TRUE based on the property .