Limits Of Trigonometric Functions

We need to find the limit:

We can rationalize the denominator by multiplying by the conjugate:

We can rewrite this as:

We know that , so we have:

Now we can substitute x = 0:

Therefore, the correct option is (A) 2.

We are given the matrix : We need to evaluate . The limit of the matrix as can be found by taking the limit of each element in the matrix. Let's evaluate the limits:
The first element is , and as , .
The second element is 1, which is constant.
The third element is 0, which is constant.
The fourth element is 0, which is constant.
The fifth element is , and as , .
The sixth element is 3, which is constant.
The seventh element is 0, which is constant.
The eighth element is 1, which is constant.
The ninth element is , and as , .
Thus, the limit of the matrix as is:

Therefore, the answer is , corresponding to option (A).