Selecting Procedures for Determining Limits - AP Calculus BC

1. What is the limit of $\text{ln}(x)$ as $x$ approaches 0 from the right?

   $-∞$. Natural logarithm approaches negative infinity at zero.

2. Find the limit: $\text{lim}_{x \to 0} \frac{e^x - 1}{x}$.

   1. This is the derivative of $e^x$ at $x = 0$.

3. What is the squeeze theorem used for?

   Finding limits of bounded functions. Traps function between two converging bounds.

4. What is the limit of $\text{e}^x$ as $x$ approaches negative infinity?

   0. Exponential function approaches zero for large negative values.

5. Find the limit: $\text{lim}_{x \to \text{∞}} \frac{5x^3 - x}{2x^3 + 3x^2}$.

   $\frac{5}{2}$. Divide by $x^3$: leading coefficients give $\frac{5}{2}$.

6. Which method is used for limits of the form $\frac{0}{0}$?

   L'Hôpital's Rule. Standard method for $\frac{0}{0}$ indeterminate forms.

7. What is the limit of $(1+\frac{1}{n})^n$ as $n$ approaches infinity?

   $e$. This is the definition of Euler's number $e$.

8. What is the limit of $\frac{x}{x+1}$ as $x$ approaches infinity?

   1. Divide by $x$: $\frac{1}{1 + \frac{1}{x}} \to \frac{1}{1} = 1$.

9. What is the limit of a constant $c$ as $x$ approaches any value?

   $c$. Constants remain unchanged regardless of variable behavior.

10. What is the limit of $\frac{1}{x}$ as $x$ approaches 0 from the positive side?

    $+\text{∞}$. Function approaches positive infinity as denominator approaches zero.

11. Identify the limit of $\frac{\text{ln}(x)}{x}$ as $x$ approaches infinity.

    0. Exponential growth dominates logarithmic growth.

12. What technique is used for limits at infinity involving polynomials?

    Divide by the highest power. Standard technique for rational functions at infinity.

13. Find the limit: $\text{lim}_{x \to -\text{∞}} \frac{x^2 + 2x + 1}{x^2}$.

    1. Divide by $x^2$: $1 + \frac{2}{x} + \frac{1}{x^2}$ where terms $\to 0$.

14. Find the limit: $\text{lim}_{x \to \text{∞}} \frac{3x + 2}{2x + 5}$.

    $\frac{3}{2}$. Divide by highest power: $\frac{3}{2} + \frac{\text{terms} \to 0}{\text{terms} \to 0}$.

15. What is the method for evaluating $\text{lim}_{x \to \text{∞}} f(x)$ when $f(x)$ is a rational function?

    Divide by the highest power. Divide numerator and denominator by highest degree term.

16. What is the limit of $\frac{x-1}{x^2-1}$ as $x$ approaches 1?

    $\frac{1}{2}$. Factor and simplify: $\frac{x-1}{(x+1)(x-1)} = \frac{1}{x+1}$.

17. What is the limit of $\frac{\text{sin}(x)}{x}$ as $x$ approaches infinity?

    0. Sine oscillates between -1 and 1 while $x$ grows.

18. What is the limit of $\frac{1 - \text{cos}(x)}{x^2}$ as $x$ approaches 0?

    $\frac{1}{2}$. Standard limit using half-angle trigonometric identity.

19. What is the limit of $\frac{\text{tan}(x)}{x}$ as $x$ approaches 0?

    1. Standard trigonometric limit equivalent to $\frac{\sin(x)}{x}$.

20. What is the limit of $\ln(x)$ as $x$ approaches 0 from the right?

    $-∞$. Natural logarithm approaches negative infinity at zero.

21. What is the limit of $\frac{\text{sin}(x)}{x}$ as $x$ approaches 0?

    1. This is a fundamental trigonometric limit.

22. What is the limit of $\frac{x^2}{e^x}$ as $x$ approaches infinity?

    0. Exponential growth dominates polynomial growth.

23. Find the limit: $\text{lim}_{x \to 0} \frac{\text{tan}(x)}{x}$.

    1. Equivalent to $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$.

24. Find the limit: $\lim_{x \to \infty} \frac{3x + 2}{2x + 5}$.

    $\frac{3}{2}$. Divide by highest power: $\frac{3}{2} + \frac{\text{terms} \to 0}{\text{terms} \to 0}$.

25. Find the limit: $\text{lim}_{x \to \frac{\text{π}}{2}} \text{tan}(x)$.

    Does not exist. Tangent has vertical asymptotes at odd multiples of $\frac{\pi}{2}$.

26. Identify the limit of $\frac{x^3}{\text{e}^x}$ as $x$ approaches infinity.

    0. Exponential growth dominates polynomial growth.

27. What is the limit of $\frac{1}{x}$ as $x$ approaches 0 from the left?

    $-\text{∞}$. Function approaches negative infinity as denominator approaches zero.

28. Find the limit: $\text{lim}_{x \to 0} \frac{\text{sin}(2x)}{x}$.

    2. Use $\lim_{x \to 0} \frac{\sin(u)}{u} = 1$ with $u = 2x$.

29. Find the limit: $\text{lim}_{x \to 0} \frac{\text{e}^x - 1}{x}$.

    1. This equals the derivative of $e^x$ at $x = 0$.

30. Which rule is used when both the numerator and denominator approach 0?

    L'Hôpital's Rule. Applies when limit has indeterminate form $\frac{0}{0}$ or $\frac{\infty}{\infty}$.

31. Find the limit: $\text{lim}_{x \to 3} \frac{x^2 - 9}{x - 3}$.

    6. Factor difference of squares: $\frac{(x+3)(x-3)}{x-3} = x+3$.

32. What is the limit of $e^x$ as $x$ approaches negative infinity?

    0. Exponential function approaches zero for large negative values.

33. Find the limit: $\lim_{x \to 0} \frac{\sin(2x)}{x}$.

    2. Apply $\lim_{x \to 0} \frac{\sin(u)}{u} = 1$ with $u = 2x$.

34. What is the limit of $x^n$ as $x$ approaches 0 for $n > 0$?

    0. Any positive power of zero equals zero.

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