Limits at Infinity and Horizontal Asymptotes - AP Calculus BC

1. Identify the horizontal asymptote of $f(x) = \frac{x^2}{x^2 - x + 1}$.

   $y = 1$. Same degree quadratics: $\frac{1}{1} = 1$.

2. What is the horizontal asymptote of $f(x) = \frac{x^3 + 5}{4x^4 + 1}$?

   $y = 0$. Numerator degree less than denominator degree.

3. What is the horizontal asymptote of $f(x) = \frac{5x}{x^2 + x}$?

   $y = 0$. Linear over quadratic approaches 0.

4. State the horizontal asymptote for $f(x) = \frac{x^2 + 2x}{x^2 - x}$.

   $y = 1$. Equal-degree quadratics: $\frac{1}{1} = 1$.

5. Identify the horizontal asymptote of $f(x) = \frac{x}{x^3 + 1}$.

   $y = 0$. Linear over cubic approaches 0.

6. Find the horizontal asymptote for $f(x) = \frac{4x^2 - x}{x^2 + x}$.

   $y = 4$. Quadratic coefficients: $\frac{4}{1} = 4$.

7. What is the horizontal asymptote of $f(x) = \frac{5x^3}{2x^2 + x}$?

   None. Numerator degree exceeds denominator degree.

8. Determine the horizontal asymptote of $f(x) = \frac{7x^3}{3x^3 + 2x}$.

   $y = \frac{7}{3}$. Cubic terms: $\frac{7}{3}$.

9. Determine the horizontal asymptote of $f(x) = \frac{3x^2 + 5}{4x^2 + 7}$.

   $y = \frac{3}{4}$. Quadratic coefficients: $\frac{3}{4}$.

10. State the horizontal asymptote for $f(x) = \frac{x^2 - 4}{x^2 + 5}$.

    $y = 1$. Equal-degree quadratics: $\frac{1}{1} = 1$.

11. What is the horizontal asymptote of $f(x) = \frac{2x}{x^2 + 1}$?

    $y = 0$. Linear over quadratic approaches 0.

12. State the horizontal asymptote for $f(x) = \frac{3x - 4}{x + 5}$.

    $y = 3$. Linear terms give ratio $\frac{3}{1} = 3$.

13. What is the limit of $f(x) = \frac{4x^2 - x + 6}{x^2 + x - 12}$ as $x$ approaches infinity?

    $4$. Leading coefficient ratio when degrees match.

14. What is the horizontal asymptote of $f(x) = \frac{2x^2 + x}{4x^2 + 3}$?

    $y = \frac{1}{2}$. Quadratic coefficients: $\frac{2}{4} = \frac{1}{2}$.

15. Determine the horizontal asymptote of $f(x) = \frac{4x}{x^3 + 1}$.

    $y = 0$. Numerator degree less than denominator degree.

16. Find the horizontal asymptote for $f(x) = \frac{4x^3 + 1}{x^3 - 2}$.

    $y = 4$. Cubic terms give $\frac{4}{1} = 4$.

17. What is the limit of $f(x) = \frac{6x^3}{3x^4 + 2}$ as $x$ approaches infinity?

    $0$. Numerator degree less than denominator degree.

18. Find the horizontal asymptote for $f(x) = \frac{3x^4 - 2x}{2x^4 + 5}$.

    $y = \frac{3}{2}$. Fourth-degree leading coefficients: $\frac{3}{2}$.

19. State the horizontal asymptote for $f(x) = \frac{2x}{x^3 + 5}$.

    $y = 0$. Linear over cubic approaches 0.

20. Identify the horizontal asymptote of $f(x) = \frac{x^2 + 3}{2x^2 - 1}$.

    $y = \frac{1}{2}$. Quadratic coefficients: $\frac{1}{2}$.

21. What is the horizontal asymptote of $f(x) = \frac{2x^2 + 4x}{x^2}$?

    $y = 2$. Simplify to $\frac{2x^2 + 4x}{x^2} = 2 + \frac{4}{x}$.

22. Identify the horizontal asymptote of $f(x) = \frac{3x^2}{x^2 + 8}$.

    $y = 3$. Same-degree quadratics: $\frac{3}{1} = 3$.

23. What is the horizontal asymptote of $f(x) = \frac{5x^3}{2x^3 + 7}$?

    $y = \frac{5}{2}$. Ratio of leading coefficients for equal-degree polynomials.

24. State the horizontal asymptote of $f(x) = \frac{x^2 + 2}{2x^2 - 3}$.

    $y = \frac{1}{2}$. Leading coefficient ratio: $\frac{1}{2}$.

25. What is the horizontal asymptote of $f(x) = \frac{2x^3}{x^2 + x}$?

    None. Numerator degree exceeds denominator degree.

26. State the horizontal asymptote for $f(x) = \frac{x^3 - x}{3x^3 + 5}$.

    $y = \frac{1}{3}$. Cubic leading coefficients: $\frac{1}{3}$.

27. What is the horizontal asymptote of $f(x) = \frac{x^3 + 2}{4x^3 + x}$?

    $y = \frac{1}{4}$. Cubic coefficients: $\frac{1}{4}$.

28. Determine the horizontal asymptote of $f(x) = \frac{3x^2 + 2}{x^2 - x}$.

    $y = 3$. Quadratic terms: $\frac{3}{1} = 3$.

29. What is the limit of $f(x) = \frac{3x^2}{2x^2 + 1}$ as $x$ approaches infinity?

    $\frac{3}{2}$. Divide leading coefficients when degrees are equal.

30. What is the horizontal asymptote of $f(x) = \frac{x^2 + 7x}{3x^2 - 9}$?

    $y = \frac{1}{3}$. Quadratic leading coefficients: $\frac{1}{3}$.

31. Find the horizontal asymptote for $f(x) = \frac{2x^4 + 3x}{x^4 + x^2}$.

    $y = 2$. Fourth-degree terms dominate: $\frac{2}{1} = 2$.

32. Identify the horizontal asymptote of $f(x) = \frac{7x^3 + 5}{3x^3 - 2}$.

    $y = \frac{7}{3}$. Equal degree cubic polynomials: $\frac{7}{3}$.

33. What is the horizontal asymptote of $f(x) = \frac{4x^4}{x^3 + 2}$?

    None. Numerator degree exceeds denominator degree.

34. What is the horizontal asymptote of $f(x) = \frac{2x^2 - 5x}{5x^2 + x}$?

    $y = \frac{2}{5}$. Quadratic terms: $\frac{2}{5}$.

35. What is the horizontal asymptote of $f(x) = \frac{4x}{x^2 + 1}$?

    $y = 0$. Linear over quadratic approaches 0.

36. Find the horizontal asymptote for $f(x) = \frac{x^3}{x^3 + x}$.

    $y = 1$. Cubic terms: $\frac{1}{1} = 1$.

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