Limits

The given function is:

We need to find the limits:

Left-hand limit:

For , we use the second part of the piecewise function, , because .

Substituting into the equation:

Thus, .

Right-hand limit:

For , we again use the second part of the piecewise function, , because .

Substituting into the equation:

Thus, .

Conclusion: The left-hand limit and right-hand limit are both . Therefore, the correct answer is:

(-4, 3)

We are given the limit:

We can simplify the expression:

As , , so:

Now, taking the limit as , we get:

Answer:

We are given the function:

We are asked to find the value of such that the function is differentiable at .

For the function to be differentiable at , it must be both continuous and have a matching derivative from both sides at .

1. Continuity Condition:

For continuity at , we need:

.

For , , so:

.

For , , so for continuity at , we must have:

,

which simplifies to:

,

so:

.

2. Differentiability Condition:

For differentiability at , we need:

.

For , , so:

.

For , , so:

For differentiability at , we must have:

.

Thus, the value of that satisfies both the continuity and differentiability conditions is:

.

The correct answer is e.

We are given the function with two different expressions for and . We are asked to find the limit as approaches from the right (i.e., ). For , we use the second expression: Now, substitute into this expression to find the limit: We know that is a constant value, and after evaluating the cosine, we find:

The correct option is (A) :