Explanation:

Let’s break down the problem bit by bit:

- We have the function f'(x) as a 3x3 matrix. The problem asks for the limit of f'(x)/x² as x approaches 0.

- First, find the determinant of the matrix, since f'(x) seems to refer to the determinant:

| cos x    x     1    |

| 2sin x  x²   2x   |

| tan x     x    1    |

- At x = 0:

- cos(0) = 1

- sin(0) = 0

- tan(0) = 0

- x terms become 0

Plugging these in, the matrix is:

| 1    0    1 |

| 0    0    0 |

| 0    0    1 |

The determinant of this becomes 0 (because the second row is all zeros).

- For the limit as x?0:

- Expand the determinant up to the order x².

- For small x, cos x ˜ 1 - x²/2, sin x ˜ x, tan x ˜ x.

- Calculate the determinant using these approximations (this is kind of messy, but you can check row by row).

- When you do the calculation, you’ll see:

- The determinant is roughly -x² as x?0.

- So, det(f'(x))/x² ˜ (-x²)/x² = -1 as x?0.

Options:

1. -1  (This is the correct answer. The limit is -1.)

2. 0

3. 1

4. 2

That’s the answer. The limit is -1, and option 1 is correct. Nothing fancy, just careful expansion and plugging things in.