Selecting Procedures for Determining Limits - AP Calculus AB

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

   2. Factor numerator: $\frac{(x+1)(x-1)}{x-1} = x+1$, so limit is $1+1=2$.

2. What is the limit of $\text{ln}(x)$ as $x$ approaches Infinity?

   Infinity. Logarithm grows without bound as argument increases.

3. Evaluate the limit: $\lim_{x \to \infty} \frac{2x + 3}{5x - 4}$.

   $\frac{2}{5}$. Divide coefficients of highest powers: $\frac{2}{5}$.

4. What is the limit of $\text{e}^x - 1$ as $x$ approaches 0?

   0. Direct substitution: $e^0 - 1 = 1 - 1 = 0$.

5. State the definition of continuity at a point.

   A function $f$ is continuous at $x = c$ if $\text{lim}_{x \to c} f(x) = f(c)$. Function is continuous when limit equals function value.

6. What is the limit of $f(x) = \frac{1}{x}$ as $x$ approaches Infinity?

   0. As denominator grows without bound, fraction approaches 0.

7. State the result of evaluating $\text{lim}_{x \to \text{Infinity}} \frac{x^2 + 3x + 2}{5x^2 - 4}$.

   $\frac{1}{5}$. Divide by highest power: coefficients of $x^2$ give $\frac{1}{5}$.

8. State the condition for a function to be continuous at $x = c$.

   $\lim_{x \to c} f(x) = f(c)$. Limit must equal function value for continuity.

9. What is the limit of $f(x) = 2x^3 - x + 5$ as $x$ approaches 0?

   5. Direct substitution: $2(0)^3 - 0 + 5 = 5$.

10. Which rule applies when directly substituting $x = c$ in rational functions?

    Direct Substitution Rule. When function is continuous at $c$, simply substitute $x=c$.

11. What is the limit of $1/x$ as $x$ approaches 0 from the right?

    Infinity. As $x \to 0^+$, denominator approaches 0 positively.

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

    0. Direct substitution: $\sin(0) = 0$.

13. What is the limit of $f(x) = 3x + 5$ as $x$ approaches 2?

    11. Direct substitution: $3(2) + 5 = 11$.

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

    6. Factor: $\frac{(x+3)(x-3)}{x-3} = x+3$, so limit is $3+3=6$.

15. What is the squeeze theorem used for?

    Determining limits of functions trapped between two other functions. If $g(x) \leq f(x) \leq h(x)$ and $\lim g(x) = \lim h(x) = L$, then $\lim f(x) = L$.

16. What is the limit of $f(x) = \frac{1}{x}$ as $x$ approaches $\infty$?

    0. As denominator grows without bound, fraction approaches 0.

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

    -Infinity. As $x \to 0^-$, denominator approaches 0 negatively.

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

    -Infinity. Logarithm approaches $-\infty$ as argument approaches 0.

19. Find the limit: $\text{lim}_{x \to -\text{Infinity}} (5x^2 + 3x - 2)$.

    Infinity. Highest power dominates: $5x^2$ term grows without bound.

20. What is the limit of $\text{cos}(x)$ as $x$ approaches $\frac{\text{pi}}{2}$?

    0. Direct substitution: $\cos(\frac{\pi}{2}) = 0$.

21. State the condition for using L'Hôpital's Rule.

    Indeterminate forms $\frac{0}{0}$ or $\frac{\text{Infinity}}{\text{Infinity}}$. Rule applies only to these specific indeterminate forms.

22. Evaluate the limit: $\text{lim}_{x \to \text{Infinity}} (\text{e}^{-x})$.

    0. Exponential decay: $e^{-x} \to 0$ as $x \to \infty$.

23. Which method is used for limits of rational functions with complex roots?

    Factor and Cancel. Factor numerator and denominator, then cancel common factors.

24. What is the limit of $\frac{\text{ln}(x)}{x}$ as $x$ approaches Infinity?

    0. L'Hôpital's Rule: $\lim_{x \to \infty} \frac{\ln(x)}{x} = \lim_{x \to \infty} \frac{1/x}{1} = 0$.

25. Determine the limit: $\text{lim}_{x \to 1} \frac{2x - 2}{x^2 - 1}$.

    1. Factor denominator: $\frac{2(x-1)}{(x+1)(x-1)} = \frac{2}{x+1}$, so limit is $\frac{2}{2}=1$.

26. What is the limit of $x^2$ as $x$ approaches -3?

    9. Direct substitution: $(-3)^2 = 9$.

27. Determine the limit: $\text{lim}_{x \to 2} \frac{x^2 - 4}{x - 2}$.

    4. Factor: $\frac{(x+2)(x-2)}{x-2} = x+2$, so limit is $2+2=4$.

28. What is the limit of $\text{tan}(x)$ as $x$ approaches $\frac{\text{pi}}{4}$?

    1. Direct substitution: $\tan(\frac{\pi}{4}) = 1$.

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

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

30. Find the limit: $\text{lim}_{x \to \text{Infinity}} (3 - \frac{5}{x})$.

    3. As $x \to \infty$, $\frac{5}{x} \to 0$, so limit is $3-0=3$.

31. Which theorem states that if $f$ is continuous on $[a, b]$, then $f$ takes every value between $f(a)$ and $f(b)$?

    Intermediate Value Theorem. Guarantees continuous functions achieve all intermediate values.

32. What is the limit of $e^x$ as $x$ approaches 0?

    1. Direct substitution: $e^0 = 1$.

33. What is the limit of $f(x) = \frac{x^2 - 1}{x^2 + 1}$ as $x$ approaches Infinity?

    1. Divide by highest power: $\frac{x^2}{x^2} = 1$ as $x \to \infty$.

34. Identify the procedure for limits with polynomials.

    Use Direct Substitution if possible. Polynomials are continuous everywhere, so substitute directly.

35. State the L'Hôpital's Rule.

    If $\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{0}{0}$, then $\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}$. Apply when limit gives $\frac{0}{0}$ or $\frac{\infty}{\infty}$ form.

36. Find the limit: $\text{lim}_{x \to 1} (3x^2 + 2x - 1)$.

    4. Direct substitution: $3(1)^2 + 2(1) - 1 = 4$.

37. Evaluate the limit: $\text{lim}_{x \to 0} \frac{1 - \text{cos}(x)}{x^2}$.

    $\frac{1}{2}$. Standard limit: $\lim_{x \to 0} \frac{1-\cos(x)}{x^2} = \frac{1}{2}$.

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