Algebraic Properties of Limits - AP Calculus AB

1. Find $\text{lim}_{x \to -2} \frac{x^2 + 4x + 4}{x + 2}$. Use algebraic properties.

   The limit is $-4$. Factor perfect square: $\frac{(x+2)^2}{x+2} = x+2$, then substitute.

2. What is the limit property for a constant multiple: $\text{If }\text{lim}_{x \to a} f(x) = L$, what is $\text{lim}_{x \to a} [cf(x)]$?

   The limit is $cL$. Constant multiple property: factor out constants.

3. Find $\text{lim}_{x \to 4} \frac{x^2 - 16}{x - 4}$. Use algebraic properties.

   The limit is $8$. Factor difference of squares: $\frac{(x+4)(x-4)}{x-4} = x+4$.

4. What is the limit of a power function: $\text{If }\lim_{x \to a} f(x) = L$, what is $\lim_{x \to a} [f(x)]^n$?

   The limit is $L^n$. Power property: raise the limit to the same power.

5. Find $\text{lim}_{x \to -1} (x^3 + 4x^2 + x - 6)$. Use algebraic properties.

   The limit is $-2$. Direct substitution: $(-1)^3 + 4(-1)^2 + (-1) - 6 = -2$.

6. What is the limit of a power function: $\text{If }\text{lim}_{x \to a} f(x) = L$, what is $\text{lim}_{x \to a} [f(x)]^n$?

   The limit is $L^n$. Power property: raise the limit to the same power.

7. Find $\text{lim}_{x \to 1} (3x^3 - 2x^2 + x - 5)$. Use algebraic properties.

   The limit is $-3$. Direct substitution: $3(1)^3 - 2(1)^2 + 1 - 5 = -3$.

8. Find $\text{lim}_{x \to 2} \frac{x^2 - 5x + 6}{x - 2}$. Use algebraic properties.

   The limit is $-1$. Factor: $\frac{(x-2)(x-3)}{x-2} = x-3$, then substitute.

9. Find $\text{lim}_{x \to 2} \frac{x^2 - 4}{x - 2}$. Use algebraic properties.

   The limit is $4$. Factor and cancel: $\frac{(x+2)(x-2)}{x-2} = x+2$, then substitute.

10. Find $\text{lim}_{x \to 0} \frac{x}{\text{sin}(x)}$. Use known results.

    The limit is $1$. Reciprocal of the standard limit $\sin(x)/x = 1$.

11. If $\text{lim}_{x \to a} f(x) = L$, what is $\text{lim}_{x \to a} (f(x) + c)$?

    The limit is $L + c$. Sum property: limit of sum equals sum of limits.

12. What is the limit of $f(x) = x$ as $x$ approaches $a$?

    The limit is $a$. The identity function equals its input value.

13. Find $\text{lim}_{x \to 4} \frac{x - 4}{\text{sqrt}(x) - 2}$. Use algebraic properties.

    The limit is $4$. Rationalize by multiplying by $\frac{\sqrt{x}+2}{\sqrt{x}+2}$.

14. Find $\text{lim}_{x \to 1} \frac{x^3 - 1}{x - 1}$. Use algebraic properties.

    The limit is $3$. Factor difference of cubes: $\frac{(x-1)(x^2+x+1)}{x-1} = x^2+x+1$.

15. Find $\text{lim}_{x \to 1} \frac{x^2 - 1}{x^2 + x - 2}$. Use algebraic properties.

    The limit is $0$. Factor both: $\frac{(x-1)(x+1)}{(x-1)(x+2)} = \frac{x+1}{x+2}$, then substitute.

16. Find $\text{lim}_{x \to 0} \frac{\text{sin}(2x)}{x}$. Use known results.

    The limit is $2$. Use $\sin(2x) = 2\sin(x)\cos(x)$ and known limit $\sin(x)/x = 1$.

17. Find $\text{lim}_{x \to 2} \frac{x^3 - 8}{x - 2}$. Use algebraic properties.

    The limit is $12$. Factor difference of cubes: $\frac{(x-2)(x^2+2x+4)}{x-2} = x^2+2x+4$.

18. Find $\text{lim}_{x \to 0} x^2 \text{cos}(x)$. Use algebraic properties.

    The limit is $0$. Product of limits: $\text{lim } x^2 \times \text{lim } \text{cos}(x) = 0 \times 1$.

19. Find $\text{lim}_{x \to 0} \frac{\text{sin}(3x)}{\text{sin}(x)}$. Use algebraic properties.

    The limit is $3$. Rewrite as $\frac{\sin(3x)}{3x} \cdot \frac{3x}{x} \cdot \frac{x}{\sin(x)} = 1 \cdot 3 \cdot 1$.

20. Find $\text{lim}_{x \to 2} \frac{\text{sqrt}(x+2) - 2}{x - 2}$. Use algebraic properties.

    The limit is $\frac{1}{4}$. Rationalize by multiplying by $\frac{\sqrt{x+2}+2}{\sqrt{x+2}+2}$.

21. What is the limit of a constant function $f(x) = c$ as $x$ approaches any value $a$?

    The limit is $c$. Constants remain unchanged regardless of the input value.

22. Find $\text{lim}_{x \to 1} \frac{x^2 - 1}{x - 1}$. Use algebraic properties.

    The limit is $2$. Factor difference of squares: $\frac{(x+1)(x-1)}{x-1} = x+1$.

23. What is the algebraic property of limits for the absolute value function?

    $\lim_{x \to a} |f(x)| = |\lim_{x \to a} f(x)|$. Absolute value of the limit equals limit of absolute value.

24. Find $\text{lim}_{x \to 3} \frac{x^2 - 9}{x - 3}$. Use algebraic properties.

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

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