Algebraic Properties of Limits - AP Calculus BC

1. Find the limit: $\text{lim}_{x \to \text{pi}} \text{sin}(x)$.

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

2. Evaluate: $\text{lim}_{x \to 1} \frac{x^3 - 1}{x - 1}$.

   3. Factor as $(x^2-1)/(x-1) = x+1$, then substitute $x=1$.

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

   1. $e^0 = 1$ by definition of exponential functions.

4. Find the limit of $\frac{x^2 - 1}{x - 1}$ as $x$ approaches 1.

   2. Factor as $(x-1)(x+1)/(x-1) = x+1$, then substitute.

5. What is the limit of $x^n$ as $x$ approaches $a$?

   $a^n$. Power functions are continuous, so direct substitution works.

6. Evaluate: $\text{lim}_{x \to 0} (x^4 + 3x^3 - 2x)$.

   0. Direct substitution: $0 + 0 - 0 = 0$.

7. Evaluate: $\text{lim}_{x \to 0} \frac{x^3}{x^2}$.

   0. Simplify to $\frac{x^3}{x^2} = x$, then substitute $x=0$.

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

   1. This is a fundamental trigonometric limit identity.

9. Find the limit of $\frac{3x^2 - x}{2x^2 + x}$ as $x$ approaches infinity.

   $\frac{3}{2}$. For rational functions, the limit equals the ratio of leading coefficients.

10. Evaluate: $\lim_{x \to 0} \frac{(1+x)^n - 1}{x}$ for $n > 0$.

    $n$. This follows from the binomial theorem and L'Hôpital's rule.

11. Evaluate: $\text{lim}_{x \to 1} (x^3 - 3x^2 + 3x - 1)$.

    0. Direct substitution: $1 - 3 + 3 - 1 = 0$.

12. Evaluate: $\text{lim}_{x \to \frac{\text{pi}}{2}} \text{sin}(x)$.

    1. Direct substitution: $\sin(\pi/2) = 1$.

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

    0. Direct substitution: $(-1)^2 + 2(-1) + 1 = 1 - 2 + 1 = 0$.

14. State the property for the limit of a sum: $\text{lim}_{x \to a} (f(x) + g(x))$.

    $\text{lim}_{x \to a} f(x) + \text{lim}_{x \to a} g(x)$. The limit of a sum equals the sum of the individual limits.

15. Evaluate: $\text{lim}_{x \to 0} \frac{\tan(x)}{x}$.

    1. Since $\tan(x) = \sin(x)/\cos(x)$ and $\cos(0) = 1$.

16. Evaluate: $\text{lim}_{x \to -2} \frac{x^2 - 4}{x + 2}$.

    -4. Factor as $(x-2)(x+2)/(x+2) = x-2$, then substitute.

17. Evaluate: $\lim_{x \to -2} \frac{x^2 - 4}{x + 2}$.

    -4. Factor as $ (x-2)(x+2)/(x+2) = x-2 $, then substitute.

18. Find the limit: $\text{lim}_{x \to 3}(5x^2 - 2x + 1)$.

    40. Direct substitution: $5(9) - 2(3) + 1 = 45 - 6 + 1 = 40$.

19. State the property for the limit of a quotient: $\lim_{x \to a} \frac{f(x)}{g(x)}$

    $\frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$, if $\lim_{x \to a} g(x) \neq 0$. The limit of a quotient equals the quotient of limits when denominator is nonzero.

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

    2. Factor as $(x-1)(x+1)/(x-1) = x+1$, then substitute.

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

    0. As $x$ grows large, $\frac{1}{x}$ approaches zero.

22. Evaluate: $\lim_{x \to 0} (x^4 + 3x^3 - 2x)$.

    0. Direct substitution: $0 + 0 - 0 = 0$.

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

    6. Factor as $(x-3)(x+3)/(x-3) = x+3$, then substitute.

24. Evaluate: $\text{lim}_{x \to 0} \frac{(1+x)^n - 1}{x}$ for $n > 0$.

    $n$. This follows from the binomial theorem and L'Hôpital's rule.

25. State the property for the limit of a product: $\text{lim}_{x \to a} (f(x) \times g(x))$.

    $\text{lim}_{x \to a} f(x) \times \text{lim}_{x \to a} g(x)$. The limit of a product equals the product of individual limits.

26. What is the limit of $\frac{x^2 - 3x + 2}{x - 2}$ as $x$ approaches 2?

    1. Factor as $ (x-1)(x-2)/(x-2) = x-1 $, then substitute.

27. State the property for the limit of a quotient: $\text{lim}_{x \to a} \frac{f(x)}{g(x)}$.

    $\frac{\text{lim}_{x \to a} f(x)}{\text{lim}_{x \to a} g(x)}$, if $\text{lim}_{x \to a} g(x) \neq 0$. The limit of a quotient equals the quotient of limits when denominator is nonzero.

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

    Infinity. Exponential growth dominates polynomial growth.

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

    4. Factor as $(x-2)(x+2)/(x-2) = x+2$, then substitute.

30. Which property allows splitting limits: $\lim_{x \to a} (f(x) + g(x))$?

    Limit of a sum. This refers to the algebraic property that splits sums.

31. State the property for the limit of a difference: $\text{lim}_{x \to a} (f(x) - g(x))$.

    $\text{lim}_{x \to a} f(x) - \text{lim}_{x \to a} g(x)$. The limit of a difference equals the difference of individual limits.

32. Evaluate: $\text{lim}_{x \to 0} \frac{\text{sin}(3x)}{x}$.

    3. Use $\frac{\sin(3x)}{x} = 3 \cdot \frac{\sin(3x)}{3x}$ and the fundamental limit.

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

    The limit is $c$. Constants remain unchanged regardless of the variable's value.

34. Identify the limit: $\text{lim}_{x \to a} c \times f(x)$.

    $c \times \text{lim}_{x \to a} f(x)$. Constants factor out of limit expressions.

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

    7. Direct substitution: $5(2) - 3 = 10 - 3 = 7$.

36. Evaluate: $\lim_{x \to 0} \frac{\tan(x)}{x}$.

    1. Since $\tan(x) = \sin(x)/\cos(x)$ and $\cos(0) = 1$.

37. Find the limit: $\text{lim}_{x \to 0} \frac{\text{sin}(5x)}{x}$.

    5. Use $\frac{\sin(5x)}{x} = 5 \cdot \frac{\sin(5x)}{5x}$ and the fundamental limit.

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

    0. Higher powers in denominators approach zero faster.

39. Find the limit of a polynomial $f(x)$ as $x$ approaches $a$.

    $f(a)$. Polynomials are continuous, so use direct substitution.

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

    1. This is a fundamental trigonometric limit identity.

41. What is the limit of $f(x) = x \times g(x)$ as $x$ approaches 0 and $g(x)$ is bounded?

    0. If $g(x)$ is bounded, then $x \cdot g(x) \to 0$ as $x \to 0$.

42. Find the limit: $\text{lim}_{x \to 0} (1 + x)^\frac{1}{x}$.

    $e$. This is the definition of $e$ as a limit.

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