Continuity And Differentiability

To determine the value of for the function to be continuous at , we must ensure that the left-hand limit, the right-hand limit, and the function value at are all equal.

Step 1: Calculate the left-hand limit as
For , . So:

Step 2: Calculate the right-hand limit as
For , . Hence:

Step 3: Enforce continuity at
We require: This gives:

Step 4: Solve for

Final Answer:

• Continuity at : A function is continuous at a point if the limit of as approaches from the left is equal to the limit of as approaches from the right, and both are equal to . For :
  1. As , .
  2. As , .
  3. Thus, .
  The function is continuous at because these values match.
• Differentiability at : A function is differentiable at a point if the derivative exists at that point. To determine if is differentiable at , we consider the left-hand derivative and the right-hand derivative:
  1. From the left, , and the derivative .
  2. From the right, , and the derivative .
  Because the left-hand and right-hand derivatives are not equal, the derivative does not exist at . Therefore, it is not differentiable at .

• is defined.
• .
• .

• (from branch).
• .
• .

• (from branch).
• .
• (same as for ).

• The left-hand limit exists.
• The right-hand limit exists.
• The left-hand limit, right-hand limit, and are equal.

Step 1: Understanding the Concept:

We need to find the derivative of a composite function, which requires the use of the chain rule multiple times. After finding the derivative, we will evaluate it at the specified point.

Step 2: Key Formula or Approach:

The chain rule states that if , then: We have , , and .

Step 3: Detailed Explanation:

The function is . Let's apply the chain rule to find :

Now, we evaluate this derivative at . First, let's find the values of the terms involving : So we have: Substitute these values back into the expression for the derivative: This can also be written as .

Step 4: Final Answer:

The value of the derivative at is . This matches option (A).

Step 1: Understanding the Concept:

We need to find the derivative of a composite function, which requires the use of the chain rule multiple times. After finding the derivative, we will evaluate it at the specified point.

Step 2: Key Formula or Approach:

The chain rule states that if , then: We have , , and .

Step 3: Detailed Explanation:

The function is . Let's apply the chain rule to find :

Now, we evaluate this derivative at . First, let's find the values of the terms involving : So we have: Substitute these values back into the expression for the derivative: This can also be written as .

Step 4: Final Answer:

The value of the derivative at is . This matches option (A).