If $f(x) = \begin{vmatrix} x & 1+x^2 & x^3 \\ \ln(1+x^2) & e^x & \sin x \\ \cos x & \tan x & \sin^2 x \end{vmatrix}$, then:
Step-by-Step Solution
Key Concept: Evaluate the determinant by first taking limits of each entry using standard limit identities ($\lim_{x\to 0} \sin x/x = 1$, $\lim_{x\to 0} \ln(1+x) \approx x$, etc.).
The matrix contains functions in entries: $\ln(1+x^2)$, $e^x$, $1+x^2$, $x^2$, $\sin x/x$, $\cos x$, $\tan x$, and $\sin^2 x/x$. Taking the limit as $x \to 0$ gives the matrix $\begin{pmatrix} 0 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 0 \end{pmatrix}$ with determinant $-1$.
Correct Answer: 4