Defining Limits and Using Limit Notation - AP Calculus BC

1. Identify the notation used to denote the left-hand limit.

   $\text{lim}_{x \to c^-} f(x)$. The minus sign indicates approaching from values less than $c$.

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

   $-8$. Direct substitution: $(-2)^3 = -8$.

3. What is the limit of $e^{-x}$ as $x$ approaches infinity?

   0. Exponential decay functions approach zero as the exponent becomes large.

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

   0. Tangent is continuous at zero, so direct substitution gives $\tan(0) = 0$.

5. State the condition for the existence of a finite limit of $f(x)$ as $x$ approaches $c$.

   The left-hand and right-hand limits must be equal. This ensures the limit is unique and well-defined.

6. What is the definition of a limit of a function as x approaches a constant c?

   The value that $f(x)$ approaches as $x$ approaches $c$. This describes how a function behaves near a point without requiring the function to be defined there.

7. What is the limit of $x^2 + 3x + 2$ as $x$ approaches -1?

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

8. State the basic limit property: $\text{lim}_{x \to c} [f(x) \times g(x)]$.

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

9. State the condition for applying the Squeeze Theorem.

   $g(x) \rightarrow L$, $h(x) \rightarrow L$, $g(x) \rightarrow f(x) \rightarrow h(x)$. Requires $f(x)$ to be bounded between $g(x)$ and $h(x)$ near the point.

10. What is the limit of $1/x$ as $x$ approaches negative infinity?

    0. As $x$ becomes large and negative, $\frac{1}{x}$ approaches zero from below.

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

    0. Exponential functions with positive bases approach zero as the exponent goes to $-\infty$.

12. State the limit notation for $f(x)$ as $x$ approaches $c$.

    $\text{lim}_{x \to c} f(x)$. Standard notation where $x$ approaches $c$ and we evaluate the limit of $f(x)$.

13. What is the limit of a quotient: $\text{lim}_{x \to c} \frac{f(x)}{g(x)}$?

    $\frac{\text{lim}_{x \to c} f(x)}{\text{lim}_{x \to c} g(x)}$ if $\text{lim}_{x \to c} g(x) \neq 0$. The quotient rule requires the denominator limit to be non-zero.

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

    Infinity. Exponential functions with base greater than 1 grow without bound as exponent increases.

15. What is the limit of a constant times a function: $\text{lim}_{x \to c} [k \times f(x)]$?

    $k \times \text{lim}_{x \to c} f(x)$. Constants factor out of limits when the limit of the function exists.

16. What does it mean for a limit to be infinite?

    The function grows without bound as $x$ approaches a value. The function increases or decreases without bound near the specified point.

17. State the limit property for powers: $\text{lim}_{x \to c} [f(x)]^n$.

    $[\text{lim}_{x \to c} f(x)]^n$. The limit of a power equals the power of the limit when the limit exists.

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

    $\text{lim}_{x \to 0^+} \frac{1}{x} = +\text{inf}$. Approaching zero from positive values makes the fraction arbitrarily large and positive.

19. Identify the notation used to denote the right-hand limit.

    $\text{lim}_{x \to c^+} f(x)$. The plus sign indicates approaching from values greater than $c$.

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

    1. This is a fundamental trigonometric limit used in derivative calculations.

21. What is the limit of $x^4$ as $x$ approaches 2?

    16. Direct substitution: $2^4 = 16$.

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

    2. Factor and cancel: $\frac{(x-1)(x+1)}{x-1} = x+1$, then substitute $x = 1$.

23. State the basic limit property: $\text{lim}_{x \to c} [f(x) + g(x)]$.

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

24. State the limit property for a function divided by a constant.

    $\frac{\text{lim}_{x \to c} f(x)}{k}$. Dividing by a non-zero constant is equivalent to multiplying by $\frac{1}{k}$.

25. State the Squeeze Theorem in limit notation.

    If $g(x) \rightarrow L$ and $h(x) \rightarrow L$, then $f(x) \rightarrow L$. If $f(x)$ is squeezed between two functions with the same limit, $f(x)$ has that limit.

26. What is the limit of a constant function $f(x) = k$ as $x$ approaches $c$?

    $k$. Constant functions have the same value everywhere, so the limit equals the constant.

27. What does it mean if the left-hand and right-hand limits are equal?

    The limit of $f(x)$ at $x = c$ exists. When both one-sided limits equal the same value, the two-sided limit exists.

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

    0. As $x$ grows large, $\frac{1}{x}$ becomes arbitrarily small.

29. State the limit of a sum of two functions: $\text{lim}_{x \to c} [f(x) + g(x)]$.

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

30. What is the limit of $x$ as $x$ approaches 5?

    5. Direct substitution works since $f(x) = x$ is continuous everywhere.

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

    1. Cosine is continuous at zero, so direct substitution gives $\cos(0) = 1$.

32. State the limit of a product of two functions: $\text{lim}_{x \to c} [f(x) \times g(x)]$.

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

33. State the limit of $\frac{\text{sin}(x)}{x}$ as $x$ approaches infinity.

    0. The sine function oscillates between -1 and 1, so $\frac{\sin(x)}{x} \to 0$ as $x \to \infty$.

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

    $\text{lim}_{x \to 0^-} \frac{1}{x} = -\text{inf}$. Approaching zero from negative values makes the fraction arbitrarily large and negative.

35. What is the limit of $x^3$ as $x$ approaches infinity?

    Infinity. Cubic functions with positive leading coefficient grow without bound as $x \to \infty$.

36. What is the limit of $\frac{1}{x^2}$ as $x$ approaches 0?

    Infinity. $\frac{1}{x^2}$ grows without bound as $x$ approaches zero from either side.

37. What is the limit of $x^5$ as $x$ approaches 0?

    0. Any polynomial with positive degree approaches zero when $x$ approaches zero.

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

    9. Direct substitution works since $x^2$ is continuous at $x = 3$.

39. What is the limit of $x^2$ as $x$ approaches infinity?

    Infinity. Quadratic functions grow without bound as $x$ approaches infinity.

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

    $-\text{inf}$. The natural logarithm approaches negative infinity as its argument approaches zero.

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