Continuity

We need to find the right-hand limit and left-hand limit of the function:

Right-hand limit (x approaches 0 from the right):

As , , so .

We can divide both the numerator and denominator by :

As , , so .

Left-hand limit (x approaches 0 from the left):

As , , so .

So, the right-hand limit is 1 and the left-hand limit is -1.

Therefore, the correct option is (B) 1 and -1.

If is continuous at x = 0, then we need to find the value of K.

For f(x) to be continuous at x = 0, we must have .

So, .

We can use the trigonometric identity :

We know that , so .

We can rewrite this as:

Since , we have:

Therefore, the correct option is (D) .

Let's evaluate both statements:

Statement 1: We are given that Substituting into the expression, we get Thus, statement 1 is true.

Statement 2: We are given that As , the denominator approaches 0, making the limit undefined. Therefore, statement 2 is false.

The correct answer is (B) : Only statement 1 is true

1. Understand the problem:

We need to analyze the continuity and differentiability of the function .

2. Analyze continuity:

The cosine function is continuous everywhere, and the absolute value function is also continuous. Therefore, is continuous everywhere.

3. Analyze differentiability:

The absolute value function is not differentiable where its argument is zero. For , this occurs at:

At these points, the left and right derivatives are not equal, making the function non-differentiable.

Correct Answer: (B) Everywhere continuous but not differentiable at odd multiples of