Limits at Infinity and Horizontal Asymptotes - AP Calculus AB

1. What is the limit as $x$ approaches infinity for $f(x) = \frac{2x + 1}{3x + 4}$?

   $\frac{2}{3}$. Same degree; ratio of leading coefficients is $\frac{2}{3}$.

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

   $y = \frac{3}{4}$. Same degree; ratio of leading coefficients is $\frac{3}{4}$.

3. Find the limit as $x$ approaches infinity for $f(x) = \frac{7x^3}{2x^3 + 5}$.

   $\frac{7}{2}$. Same degree; ratio of leading coefficients is $\frac{7}{2}$.

4. Determine $\text{lim}_{x \to \infty} \frac{5x^2 + 3}{3x^2 - 2}$.

   $\frac{5}{3}$. Same degree; ratio of leading coefficients is $\frac{5}{3}$.

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

   $y = 0$. Denominator degree exceeds numerator degree, so asymptote is $y = 0$.

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

   $\infty$. Numerator degree exceeds denominator degree, so limit approaches $\infty$.

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

   None. Numerator degree exceeds denominator degree, so no horizontal asymptote exists.

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

   $y = 0$. Denominator degree exceeds numerator degree, so asymptote is $y = 0$.

9. Determine $\text{lim}_{x \to \infty} \frac{8x - 3}{2x + 1}$.

   $4$. Same degree; ratio of leading coefficients is $\frac{8}{2} = 4$.

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

    $y = \frac{7}{2}$. Same degree polynomials; ratio of leading coefficients is $\frac{7}{2}$.

11. Find the limit as $x$ approaches infinity for $f(x) = \frac{x^2 - 1}{2x^2 + x}$.

    $\frac{1}{2}$. Same degree; ratio of leading coefficients is $\frac{1}{2}$.

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

    $y = 0$. Denominator degree exceeds numerator degree, so horizontal asymptote is $y = 0$.

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

    $y = 2$. Same degree numerator and denominator; ratio of leading coefficients is $\frac{2}{1} = 2$.

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

    $y = \frac{1}{2}$. Same degree; ratio of leading coefficients is $\frac{1}{2}$.

15. Determine $\text{lim}_{x \to -\infty} \frac{x^2 - 2x}{x^2 + x}$.

    $1$. Same degree; ratio of leading coefficients is $\frac{1}{1} = 1$.

16. What is the horizontal asymptote of $f(x) = \frac{8x^3 + 1}{4x^3 + 2}$?

    $y = 2$. Same degree; ratio of leading coefficients is $\frac{8}{4} = 2$.

17. Determine $\text{lim}_{x \to -\infty} \frac{6x - 1}{2x + 3}$.

    $3$. Same degree; ratio of leading coefficients is $\frac{6}{2} = 3$.

18. Determine $\text{lim}_{x \to \infty} \frac{2x^3 - 5}{3x^3 + 1}$.

    $\frac{2}{3}$. Same degree; ratio of leading coefficients is $\frac{2}{3}$.

19. What is the limit as $x$ approaches infinity for $f(x) = \frac{6x^2 - 9}{3x^2 + 7}$?

    $2$. Same degree; ratio of leading coefficients is $\frac{6}{3} = 2$.

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

    $y = \frac{2}{5}$. Same degree; ratio of leading coefficients is $\frac{2}{5}$.

21. Find the limit as $x$ approaches infinity of $f(x) = \frac{4x^2}{2x^2 + 3}$.

    $2$. Same degree; ratio of leading coefficients is $\frac{4}{2} = 2$.

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

    $y = \frac{7}{3}$. Same degree; ratio of leading coefficients is $\frac{7}{3}$.

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

    Infinity. Numerator degree exceeds denominator degree, so limit approaches infinity.

24. Find the limit as $x$ approaches infinity of $f(x) = \frac{3x^4}{x^4 + 5}$.

    $3$. Same degree; ratio of leading coefficients is $\frac{3}{1} = 3$.

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

    $y = 0$. Denominator degree exceeds numerator degree, so asymptote is $y = 0$.

26. Find the horizontal asymptote of $f(x) = \frac{6x^3}{3x^3 + 1}$.

    $y = 2$. Same degree; ratio of leading coefficients is $\frac{6}{3} = 2$.

27. What is the horizontal asymptote for $f(x) = \frac{9x + 7}{3x + 1}$?

    $y = 3$. Same degree; ratio of leading coefficients is $\frac{9}{3} = 3$.

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

    $y = \frac{2}{5}$. Same degree; ratio of leading coefficients is $\frac{2}{5}$.

29. What is the horizontal asymptote for $f(x) = \frac{2x^3}{5x^3 - 4}$?

    $y = \frac{2}{5}$. Same degree; ratio of leading coefficients is $\frac{2}{5}$.

30. Determine $\text{lim}_{x \to \infty} \frac{5x - 1}{2x + 3}$.

    $\frac{5}{2}$. Divide by highest power; $\frac{5}{2}$ is the ratio of leading coefficients.

31. Determine $\text{lim}_{x \to \infty} \frac{7x^2 - 5}{2x^2 + 4}$.

    $\frac{7}{2}$. Same degree; ratio of leading coefficients is $\frac{7}{2}$.

32. Determine $\lim_{x \to \infty} \frac{5x - 1}{2x + 3}$.

    $\frac{5}{2}$. Divide by highest power; $\frac{5}{2}$ is the ratio of leading coefficients.

33. What is the horizontal asymptote for $f(x) = \frac{4x}{x + 5}$?

    $y = 4$. Same degree; ratio of leading coefficients is $\frac{4}{1} = 4$.

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

    $y = 0$. Denominator degree exceeds numerator degree, so asymptote is $y = 0$.

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

    $y = \frac{5}{3}$. Same degree; ratio of leading coefficients is $\frac{5}{3}$.

36. Determine $\text{lim}_{x \to -\infty} \frac{-x^3 + 4}{3x^3 - x}$.

    $\frac{-1}{3}$. Same degree; ratio of leading coefficients is $\frac{-1}{3}$.

37. What is the horizontal asymptote of $f(x) = \frac{4x^3 + 5}{x^3 + 6}$?

    $y = 4$. Same degree; ratio of leading coefficients is $\frac{4}{1} = 4$.

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

    $y = 4$. Same degree; ratio of leading coefficients is $\frac{4}{1} = 4$.

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

    $y = 0$. Denominator degree exceeds numerator degree, so asymptote is $y = 0$.

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

    $y = 0$. Denominator degree exceeds numerator degree, so asymptote is $y = 0$.

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

    $y = \frac{5}{4}$. Same degree; ratio of leading coefficients is $\frac{5}{4}$.

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