To determine the slant asymptote of the function $f(x) = \frac{4x^2 - x - 2}{x - 1}$, we use polynomial division because a slant (oblique) asymptote occurs when the degree of the numerator is exactly one higher than the degree of the denominator, as it is here.

Steps:

1. Perform Polynomial Long Division: Divide $4x^2 - x - 2$ by $x - 1$.

   • First, divide $4x^2$ by $x$, which gives $4x$.

   • Multiply $4x$ by $x - 1$, which gives $4x^2 - 4x$.

   • Subtract $4x^2 - 4x$ from $4x^2 - x - 2$, yielding $3x - 2$.

   • Next, divide $3x$ by $x$, which gives $+3$.

   • Multiply $3$ by $x - 1$, which gives $3x - 3$.

   • Subtract $3x - 3$ from $3x - 2$, yielding a remainder of $+1$.

2. Obtain the Equation of the Slant Asymptote: The result of the division is $f(x) = 4x + 3 + \frac{1}{x - 1}$.

   As $x \to \infty$, the remainder term $\frac{1}{x - 1} \to 0$, so the slant asymptote is given by:

   $y = 4x + 3$

Thus, the slant asymptote is $y = 4x + 3$.

Would you like further details or have any questions?

Related Questions:

1. How can polynomial division be used for other rational functions?
2. What is the difference between vertical and slant asymptotes?
3. Can a rational function have both a horizontal and a slant asymptote?
4. What happens to the graph of $f(x)$ near the vertical asymptote at $x = 1$?
5. How does the remainder term $\frac{1}{x - 1}$ affect the behavior of $f(x)$?

Tip:

Always check the degree of the numerator and denominator when looking for slant asymptotes—this will guide you to the right method!