Let f(x)`=|{:(secx,x^(2),x),(2sinx,x^(3),2x^(2)),(tan3x,x^(2),x):}|lim_(x to 0)f(x)/(x^(4))` is equal to

Options

A. 0
B. -1
C. 2
D. 3

Answer

To solve the given problem, we need to evaluate the limit of the determinant as \( x \) approaches 0. Let's break down the steps:

### Step 1: Write the Determinant
The determinant is given as:
\[ f(x) = \begin{vmatrix} \sec x & x^2 & x \\ 2 \sin x & x^3 & 2x^2 \\ \tan 3x & x^2 & x \end{vmatrix} \]

### Step 2: Factor Out Common Terms
We can factor out common terms from the columns and rows of the determinant.

- From column 2 (C2), we can take out \( x^2 \).
- From column 3 (C3), we can take out \( x \).
- From row 2 (R2), we can take out \( 2x \).

After factoring, we have:
\[ f(x) = x^4 \begin{vmatrix} \sec x & 1 & 1 \\ 2 \sin x/x & x & 2 \\ \tan 3x/x & 1 & 1 \end{vmatrix} \]

### Step 3: Simplify the Determinant
Now we need to evaluate the limit:
\[ \lim_{x \to 0} \frac{f(x)}{x^4} = \lim_{x \to 0} \begin{vmatrix} \sec x & 1 & 1 \\ 2 \frac{\sin x}{x} & 1 & 2 \\ \tan 3x/x & 1 & 1 \end{vmatrix} \]

### Step 4: Evaluate the Limit
As \( x \) approaches 0:
- \( \sec(0) = 1 \)
- \( \frac{\sin x}{x} \to 1 \)
- \( \tan(3x)/x \to 3 \) (using the limit property \( \tan(kx)/x \to k \) as \( x \to 0 \))

Substituting these values into the determinant, we get:
\[ \begin{vmatrix} 1 & 1 & 1 \\ 2 & 1 & 2 \\ 3 & 1 & 1 \end{vmatrix} \]

### Step 5: Calculate the Determinant
Now, we calculate the determinant:
\[ = 1 \cdot (1 \cdot 1 - 2 \cdot 1) - 1 \cdot (2 \cdot 1 - 2 \cdot 3) + 1 \cdot (2 \cdot 1 - 1 \cdot 3) \]
\[ = 1 \cdot (1 - 2) - 1 \cdot (2 - 6) + 1 \cdot (2 - 3) \]
\[ = 1 \cdot (-1) - 1 \cdot (-4) + 1 \cdot (-1) \]
\[ = -1 + 4 - 1 = 2 \]

### Final Answer
Thus, the limit evaluates to:
\[ \lim_{x \to 0} \frac{f(x)}{x^4} = 2 \]