Find the vertical and horizontal asymptotes of R(x)=(3x^2-2x+4)/(x^2+1).

Find the vertical and horizontal asymptotes of R(x)=(3x^2-2x+4)/(x^2+1).

• A. Vertical: none; Horizontal asymptote: y=3
• B. Vertical: none; Horizontal asymptote: y=0
• C. Vertical: x=0; Horizontal asymptote: y=3
• D. Vertical: x=2; Horizontal asymptote: y=3

Answer: (A)

When a rational function has the same degree in the numerator and denominator, its end behavior is governed by the ratio of the leading coefficients. Here the leading terms are 3x^2 and x^2, so as x grows large, R(x) approaches 3. This gives a horizontal asymptote at y = 3. For vertical asymptotes, look where the denominator is zero. The denominator x^2 + 1 equals zero would require x^2 = -1, which has no real solution, so there are no vertical asymptotes. So the graph has no vertical asymptotes and a horizontal asymptote at y = 3. This matches the behavior seen by rewriting the function as 3 + (-2x + 1)/(x^2 + 1), where the remainder part tends to 0 as x → ±∞.

When a rational function has the same degree in the numerator and denominator, its end behavior is governed by the ratio of the leading coefficients. Here the leading terms are 3x^2 and x^2, so as x grows large, R(x) approaches 3. This gives a horizontal asymptote at y = 3.

For vertical asymptotes, look where the denominator is zero. The denominator x^2 + 1 equals zero would require x^2 = -1, which has no real solution, so there are no vertical asymptotes.

So the graph has no vertical asymptotes and a horizontal asymptote at y = 3. This matches the behavior seen by rewriting the function as 3 + (-2x + 1)/(x^2 + 1), where the remainder part tends to 0 as x → ±∞.