Connecting Multiple Representations of Limits - AP Calculus AB

1. What is the limit of $f(x) = \frac{\text{e}^x - 1}{x}$ as $x$ approaches 0?

   1. Standard exponential limit: derivative of $e^x$ at x=0.

2. What is the limit of $f(x) = \frac{1}{x}$ as $x$ approaches 0 from the left?

   $-\text{∞}$. From the left, x is negative, so $\frac{1}{x}$ approaches negative infinity.

3. State the condition under which $\text{lim}_{x \to c} f(x)$ exists.

   Left-hand limit equals right-hand limit. Both one-sided limits must exist and be equal.

4. State the limit of $f(x) = \frac{x^3 - 27}{x - 3}$ as $x$ approaches 3.

   27. Factor: $\frac{(x-3)(x^2+3x+9)}{x-3} = x^2+3x+9$, evaluate at x=3.

5. Identify the one-sided limit: $\text{lim}_{x \to 0^+} \frac{1}{x}$.

   $+\text{∞}$. From the right, $\frac{1}{x}$ approaches positive infinity.

6. Identify the limit of $f(x) = \frac{x^2 + 2x + 1}{x + 1}$ as $x$ approaches -1.

   0. Factor: $\frac{(x+1)^2}{x+1} = x+1$, so limit is 0 at x=-1.

7. Identify the limit of $f(x) = x^3$ as $x$ approaches $-1$.

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

8. State the limit of $f(x) = \frac{2x+1}{x+3}$ as $x$ approaches $\text{∞}$.

   2. Divide leading coefficients: $\frac{2}{1} = 2$.

9. State the limit of $f(x) = \frac{1}{x^2}$ as $x$ approaches 0 from the left.

   $+\infty$. $\frac{1}{x^2}$ is always positive and grows without bound near 0.

10. What is the graphical representation of a limit?

    The $y$-value approaches as $x$ approaches a point. The height the graph approaches as x gets close to the target value.

11. What does it mean for a limit to not exist?

    The left-hand and right-hand limits do not match. The function approaches different values from left and right sides.

12. Find the limit: $\lim_{x \to \infty} \frac{2x^3 + 3}{5x^3 + 4}$.

    $\frac{2}{5}$. Divide leading coefficients when degrees are equal: $\frac{2}{5}$.

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

    1. This is a standard trigonometric limit.

14. Identify the limit: $\text{lim}_{x \to 0} \frac{\text{cos}(x) - 1}{x}$.

    0. Use L'Hôpital's rule or the identity $\cos(x) - 1 \approx -\frac{x^2}{2}$.

15. Identify the limit: $\lim_{x \to \infty} \frac{1}{x}$.

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

16. What is the limit of $f(x) = x^2 \text{e}^{-x}$ as $x$ approaches $\text{∞}$?

    0. Exponential decay dominates polynomial growth at infinity.

17. State the condition under which $\lim_{x \to c} f(x)$ exists.

    Left-hand limit equals right-hand limit. Both one-sided limits must exist and be equal.

18. State the limit of $f(x) = \frac{x}{\text{e}^x}$ as $x$ approaches $\text{∞}$.

    0. Exponential growth dominates linear growth at infinity.

19. Which theorem guarantees the existence of a limit?

    The Squeeze Theorem. Used when a function is bounded between two functions with the same limit.

20. What is the limit of $f(x) = x^2$ as $x$ approaches $3$?

    9. Direct substitution: $3^2 = 9$.

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

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

22. Identify the limit of $f(x) = 3x + 4$ as $x$ approaches 2.

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

23. What is the graphical indicator of a limit not existing at a point?

    A jump, vertical asymptote, or different left/right limits. Discontinuities appear as breaks, holes, or vertical asymptotes in the graph.

24. What is the limit of $f(x) = \frac{x^2 - 4}{x - 2}$ as $x$ approaches 2?

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

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

    1. This is a standard trigonometric limit.

26. Identify the limit: $\lim_{x \to \infty} (1 + \frac{1}{x})^x$.

    $e$. This is the definition of the mathematical constant e.

27. State the limit of $f(x) = x^2 \text{sin}(\frac{1}{x})$ as $x$ approaches 0.

    0. Use Squeeze Theorem: $|\sin(\frac{1}{x})| \leq 1$, so $|x^2\sin(\frac{1}{x})| \leq x^2$.

28. Define a continuous function in terms of limits.

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

29. Identify the limit of $f(x) = \text{tan}(x)$ as $x$ approaches $\frac{\text{π}}{2}$ from the left.

    $+\text{∞}$. Tangent has a vertical asymptote at $\frac{\pi}{2}$, approaching $+\infty$ from the left.

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

    Does not exist. The denominator approaches 0 while numerator approaches 2, creating a vertical asymptote.

31. What is the limit of $f(x) = x \text{e}^{-x}$ as $x$ approaches $\text{∞}$?

    0. Exponential decay dominates linear growth at infinity.

32. What is the limit of $f(x) = \frac{e^x - 1}{x}$ as $x$ approaches 0?

    1. Standard exponential limit: derivative of $e^x$ at x=0.

33. What is the limit of $f(x) = \frac{\text{ln}(x+1)}{x}$ as $x$ approaches 0?

    1. Standard logarithmic limit: derivative of $\ln(x+1)$ at x=0.

34. Identify the limit: $\text{lim}_{x \to \text{∞}} (1+\frac{1}{x})^x$.

    $e$. This is the definition of the mathematical constant e.

35. What is the limit of $f(x) = \frac{\sin(2x)}{x}$ as $x$ approaches 0?

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

36. Identify the limit: $\lim_{x \to c} 5$ where $c$ is any real number.

    $5$. Constant functions have the same limit everywhere.

37. What is the limit of $f(x) = \frac{1 - \text{cos}(x)}{x^2}$ as $x$ approaches 0?

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

38. State the limit of $f(x) = \frac{1}{x}$ as $x$ approaches 0 from the right.

    $+\text{∞}$. As x approaches 0 from positive values, $\frac{1}{x}$ grows without bound.

39. Find the limit of $f(x) = \frac{x^2 - 9}{x - 3}$ as $x$ approaches 3.

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

40. What is the limit of $f(x) = \text{ln}(x)$ as $x$ approaches $\text{∞}$?

    $+\text{∞}$. Natural logarithm grows without bound as x increases.

41. What is the limit of $f(x) = e^x$ as $x$ approaches $0$?

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

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