Find all asymptotes of the rational function f(x) = (2x^3 - 5x + 1) / (x^2 - 4).
Answer and explanation
Correct answer: A. Vertical: x = 2 and x = -2; Oblique: y = 2*x
- Step 1: Vertical asymptotes occur where the denominator is zero and numerator is non-zero: x^2 - 4 = 0 gives x = 2 and x = -2.
- Step 2: Since the degree of the numerator (3) is exactly 1 greater than the degree of the denominator (2), there is an oblique asymptote.
- Step 3: Perform polynomial division: (2x^3 - 5x + 1) / (x^2 - 4) = 2x + (3x + 1)/(x^2 - 4).
- Step 4: As |x| -> infinity, the remainder approaches 0, so the oblique asymptote is y = 2*x.
Answer
Vertical: x = 2 and x = -2; Oblique: y = 2*x.
Why not the others: Horizontal: y = 2 only occurs if degrees are equal; Vertical: x = 4 misfactors x^2 - 4; Horizontal: y = 0 occurs if the numerator degree is strictly smaller.
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