Final answer:
The oblique asymptote of the function g(x) = (x² + x + 4) / (x - 1) is determined by polynomial division, resulting in the line y = x + 2. The remainder of the division does not affect the asymptote as x approaches infinity.
Step-by-step explanation:
The equation given is g(x) = (x² + x + 4) / (x - 1). To find the oblique asymptote for this rational function, we need to perform polynomial division, because the degree of the numerator is one higher than the degree of the denominator. In cases like this, the quotient of the division will provide the equation of the oblique (or slant) asymptote.
Let's perform the division:
- Divide x² by x to get x.
- Multiply (x - 1) by x to get x² - x.
- Subtract (x² - x) from (x² + x) to get 2x.
- Divide 2x by x to get 2.
- Multiply (x - 1) by 2 to get 2x - 2.
- Subtract (2x - 2) from (2x + 4) to get 6.
The quotient of our division is x + 2, and the remainder is 6, which when divided by x - 1 goes to zero as x approaches infinity. Thus, the oblique asymptote of g(x) is the line y = x + 2. The existence of a remainder does not affect the oblique asymptote since the term becomes negligible for large values of x.