Step 1: Understanding the Concept:
This problem involves finding the limit of a rational function as the variable approaches infinity. The key is to compare the degrees of the numerator and the denominator.
Step 2: Key Formula or Approach:
For a rational function $f(x) = \frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials, as $x \to \infty$:
If degree(P)<degree(Q), the limit is 0.
If degree(P)>degree(Q), the limit is $\pm\infty$.
If degree(P) = degree(Q), the limit is the ratio of the leading coefficients.
Step 3: Detailed Explanation:
The given limit is:
\[ \lim_{x\to\infty} \frac{4x^3-x+1}{x^2-4x(1-x^2)} \]
First, simplify the denominator:
\[ x^2 - 4x(1-x^2) = x^2 - 4x + 4x^3 \]
So the expression becomes:
\[ \lim_{x\to\infty} \frac{4x^3-x+1}{4x^3+x^2-4x} \]
The degree of the polynomial in the numerator is 3.
The degree of the polynomial in the denominator is 3.
Since the degrees are equal, the limit is the ratio of the coefficients of the highest power term ($x^3$).
\[ \text{Leading coefficient of numerator} = 4 \]
\[ \text{Leading coefficient of denominator} = 4 \]
The value of the limit is:
\[ \text{Limit} = \frac{4}{4} = 1 \]
Step 4: Final Answer:
The value of the limit is 1. Therefore, option (B) is correct.