Defining Limits and Using Limit Notation - AP Calculus AB

1. Evaluate $\lim_{{x \to \infty}} (2x^2 + 3x + 1)$.

   $\infty$. Polynomial with positive leading coefficient grows without bound as $x \to \infty$.

2. State the limit law for powers of a function.

   $\lim_{x \to a} [f(x)]^n = [\lim_{x \to a} f(x)]^n$. Limit of power equals power of limit when the base limit exists.

3. What is an asymptote?

   A line that a graph approaches but never touches. Describes boundary behavior where graphs get arbitrarily close but never intersect the line.

4. What does it mean if $\lim_{{x \to a}} f(x)$ does not exist?

   The left-hand and right-hand limits are not equal or at least one is undefined. Different one-sided limits or undefined behavior prevents limit existence.

5. Evaluate $\lim_{{x \to 2}} (3x + 4)$.

   10. Direct substitution: $3(2) + 4 = 6 + 4 = 10$.

6. Evaluate $\lim_{{x \to \infty}} \frac{5x^2 - 4x}{2x^2 + 3}$.

   $\frac{5}{2}$. Divide by highest power $x^2$: $\frac{5-\frac{4}{x}}{2+\frac{3}{x^2}} \to \frac{5}{2}$.

7. What is a one-sided limit?

   The limit of a function as $x$ approaches a point from one side, either the left or the right. Considers approach from only left ($a^-$) or right ($a^+$) side of the point.

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

   $c$. Constant functions have limits equal to the constant value everywhere.

9. What is the formal name for $\lim_{{x \to a^+}} f(x)$ and $\lim_{{x \to a^-}} f(x)$?

   Right-hand limit and left-hand limit, respectively. These describe approaches from the positive and negative sides respectively.

10. Identify the limit law for the constant multiple of a function.

    $\lim_{{x \to a}} [c \cdot f(x)] = c \cdot \lim_{{x \to a}} f(x)$. Constants can be factored out of limit expressions.

11. State the limit law for the difference of two functions.

    $\lim_{{x \to a}} [f(x) - g(x)] = \lim_{{x \to a}} f(x) - \lim_{{x \to a}} g(x)$. Limit of difference equals difference of limits when both individual limits exist.

12. What does continuity at a point $x = a$ require?

    $f(a)$ is defined, $\lim_{{x \to a}} f(x)$ exists, and $\lim_{{x \to a}} f(x) = f(a)$. Three conditions ensure no gaps, jumps, or undefined points.

13. Evaluate $\lim_{{x \to 3}} (x^2 - 9)$.

    0. Direct substitution: $(3)^2 - 9 = 9 - 9 = 0$.

14. Identify the limit of $\frac{1}{x}$ as $x$ approaches $0$ from the left.

    $-\infty$. As $x$ approaches $0^-$, $\frac{1}{x}$ becomes arbitrarily large and negative.

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

    $a$. The identity function $f(x) = x$ approaches the point value.

16. How is the left-hand limit of $f(x)$ as $x$ approaches $a$ denoted?

    $\lim_{{x \to a^-}} f(x)$. The $-$ superscript indicates approaching from values less than $a$.

17. How is the right-hand limit of $f(x)$ as $x$ approaches $a$ denoted?

    $\lim_{{x \to a^+}} f(x)$. The $+$ superscript indicates approaching from values greater than $a$.

18. What is a removable discontinuity?

    A discontinuity that can be removed by redefining the function at a point. The limit exists but the function isn't defined or has wrong value at that point.

19. Evaluate $\lim_{x \to \infty} \frac{1}{x}$.

    0. As $x$ grows without bound, $\frac{1}{x}$ approaches zero.

20. Evaluate $\lim_{{x \to 1}} \frac{x^2 - 1}{x - 1}$.

    2. Factor as $\frac{(x+1)(x-1)}{x-1}$, cancel to get $x+1$, then substitute $x=1$.

21. What is the condition for the existence of a two-sided limit?

    Both one-sided limits must exist and be equal. When $\lim_{x \to a^+} f(x) = \lim_{x \to a^-} f(x)$, the two-sided limit exists.

22. State the limit law for the sum of two functions.

    $\lim_{{x \to a}} [f(x) + g(x)] = \lim_{{x \to a}} f(x) + \lim_{{x \to a}} g(x)$. Limit of sum equals sum of limits when both individual limits exist.

23. Evaluate $\lim_{{x \to 0}} \frac{\sin x}{x}$.

    1. This is a fundamental trigonometric limit used in calculus.

24. State the limit law for the product of two functions.

    $\lim_{{x \to a}} [f(x) \cdot g(x)] = \lim_{{x \to a}} f(x) \cdot \lim_{{x \to a}} g(x)$. Limit of product equals product of limits when both individual limits exist.

25. Identify the limit of $\frac{1}{x}$ as $x$ approaches $0$ from the right.

    $\infty$. As $x$ approaches $0^+$, $\frac{1}{x}$ becomes arbitrarily large and positive.

26. Evaluate $\lim_{{x \to \infty}} \frac{3x^3 + x + 2}{x^3}$.

    3. Divide numerator and denominator by $x^3$; limit of $\frac{3x^3}{x^3} = 3$.

27. If $\lim_{{x \to a}} f(x) = L$, what is the relationship between $f(x)$ and $L$?

    As $x$ approaches $a$, $f(x)$ gets arbitrarily close to $L$. The function output approaches $L$ but doesn't necessarily equal $L$ at $x = a$.

28. What is the definition of a limit in calculus?

    The value a function approaches as the input approaches a point. This captures the fundamental concept of approaching a value without necessarily reaching it.

29. What is the limit notation for $f(x)$ as $x$ approaches infinity?

    $\lim_{{x \to \infty}} f(x)$. Examines function behavior as input values become arbitrarily large.

30. What is an infinite limit?

    A limit where $f(x)$ increases or decreases without bound as $x$ approaches a point. Function values grow without bound, approaching positive or negative infinity.

31. How is the limit of a function as $x$ approaches $a$ denoted?

    $\lim_{{x \to a}} f(x)$. Standard mathematical notation using $\lim$ with subscript showing the approach.

32. How is an infinite limit denoted?

    $\lim_{x \to a} f(x) = \infty$ or $-\infty$. Uses infinity symbol to show unbounded growth in positive or negative direction.

33. Evaluate $\lim_{{x \to -1}} (x^3 + 2x^2 + x)$.

    0. Direct substitution: $(-1)^3 + 2(-1)^2 + (-1) = -1 + 2 - 1 = 0$.

34. What does it mean if $\lim_{{x \to a}} f(x) = \pm \infty$?

    The function $f(x)$ has a vertical asymptote at $x = a$. Function values approach infinity, creating a vertical line the graph approaches.

35. What does it mean if $\lim_{{x \to a}} f(x) = L$?

    As $x$ approaches $a$, $f(x)$ approaches $L$. The function value gets arbitrarily close to $L$ as input approaches $a$.

36. State the limit law for the quotient of two functions.

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

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