Continuity

The correct answer is (B) :

For continuity at x = 0,
Now,

⇒ p = 3 (To make indeterminant form)
So,

To ensure continuity of at , the left-hand limit and the right-hand limit as approaches must both equal .

1. Left-Hand Limit ( ):

As , , so:

Thus,

2. Right-Hand Limit ( ):

Simplify the exponent:

As :

Applying L'Hôpital's Rule to the indeterminate form:

Therefore,

3. Continuity Condition:

Thus,

Thus, ,

4. Evaluating the Expression:

Thus, the correct answer is option (4).

To solve the problem, we need to ensure that the given function is continuous at . Continuity at this point means that the left-hand limit, the value of the function at , and the right-hand limit must all be equal.

1. Left-hand limit as :

We have the function for . To find the limit as approaches zero from the left, we use the logarithm:

The left-hand limit is .

1. Value at :

The value of the function at is given by .

1. Right-hand limit as :

For , . We need to find the limit as approaches zero from the right:

Applying L'Hospital's Rule (since both numerator and denominator approach 0), we get:

The right-hand limit is .

1. Ensure continuity:

For continuity at , set the left-hand limit equal to the value at zero and the right-hand limit:

Solve these equations simultaneously.

From , we get .

From , we get , solve to find .

Combine all to find :

Given all conditions satisfy .

Thus, the answer is 48.