Confirming Continuity over an Interval - AP Calculus AB

1. Find $\text{lim}_{x \to 2} (x^2 - 4)/(x - 2)$. Is it continuous at $x = 2$?

   Limit is $4$, not continuous without $f(2)$ defined. Use L'Hôpital's rule or factor: limit is $4$, but function undefined.

2. Find the limit: $\lim_{x \to 3} (2x + 1)$.

   $7$. Direct substitution works for polynomial functions.

3. Find $\lim_{x \to 0} \frac{\sin(x)}{x}$

   $1$. Standard trigonometric limit used in derivative of sine.

4. If $f(x) = \frac{x^2 - 4}{x - 2}$, what value must $f(2)$ be for continuity at $x = 2$?

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

5. What is a point of discontinuity?

   A point where $f(x)$ is not continuous. Where any of the three continuity conditions fail.

6. What kind of discontinuity occurs if $\text{lim}_{x \to a^-} f(x) \neq \text{lim}_{x \to a^+} f(x)$?

   A jump discontinuity. One-sided limits exist but differ, creating a vertical 'jump'.

7. What type of discontinuity is present if $f(x)$ is undefined at $x = a$?

   It could be a removable or infinite discontinuity. Depends on whether the function approaches infinity or has a finite limit.

8. Is $g(x) = \text{ln}(x)$ continuous for $x > 0$?

   Yes, $g(x)$ is continuous for $x > 0$. Natural logarithm is continuous on its entire domain.

9. What must be true about $\text{lim}_{x \to a^-} f(x)$ and $\text{lim}_{x \to a^+} f(x)$ for continuity at $x = a$?

   They must both exist and be equal to $f(a)$. Left and right limits must converge to the same value as the function.

10. Which theorem guarantees a root exists in $[a, b]$ if $f(a) \times f(b) < 0$?

    The Intermediate Value Theorem. Sign change guarantees a zero by IVT for continuous functions.

11. If $\text{lim}_{x \to a} f(x) = L$ and $f(a) = L$, is $f(x)$ continuous at $x = a$?

    Yes, $f(x)$ is continuous at $x = a$. All three conditions for continuity are satisfied.

12. What ensures a function is continuous on a closed interval $[a, b]$?

    Continuous on $(a, b)$, right at $a$, left at $b$. Requires continuity at all interior points plus endpoint conditions.

13. Determine if $f(x) = |x|$ is continuous at $x = 0$.

    Yes, $f(x)$ is continuous at $x = 0$. Left and right limits both equal $0$, matching $f(0) = |0| = 0$.

14. What is the definition of continuity at a point $x = a$?

    A function $f(x)$ is continuous at $x = a$ if $\text{lim}_{x \to a} f(x) = f(a)$. The limit must exist and equal the function value at that point.

15. What must be true for $f(x)$ to be continuous from the right at $x = a$?

    $\text{lim}_{x \to a^+} f(x) = f(a)$. Right-hand limit must equal the function value.

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

    $1$. Standard trigonometric limit used in derivative of sine.

17. What is a removable discontinuity?

    A point where a function is not defined, but the limit exists. Can be 'fixed' by defining or redefining the function at that point.

18. Which condition must be checked first to confirm continuity on an interval $[a, b]$?

    Check if $f(x)$ is defined for all $x \text{ in } [a, b]$. Domain restrictions are the most common source of discontinuities.

19. For $g(x) = \frac{x^2 - 1}{x - 1}$, does $g(x)$ have a removable discontinuity at $x = 1$?

    Yes, $g(x)$ has a removable discontinuity. Factor: $\frac{(x+1)(x-1)}{x-1} = x+1$, limit exists at $x = 1$.

20. What is the limit $\text{lim}_{x \to 2} 3x + 1$?

    $7$. Direct substitution: $3(2) + 1 = 7$.

21. Is $h(x) = \frac{1}{x-1}$ continuous at $x = 1$?

    No, $h(x)$ is undefined at $x = 1$. Vertical asymptote creates an infinite discontinuity.

22. If $f(x) = x^3$, is $f(x)$ continuous on $(-\text{\textinfty}, \text{\textinfty})$?

    Yes, $f(x)$ is continuous everywhere. Polynomial functions are continuous everywhere in their domain.

23. What is the first step in confirming continuity at a point?

    Verify $f(a)$ is defined. Cannot check continuity if the function value doesn't exist.

24. Is $f(x) = \frac{1}{x}$ continuous at $x = 0$? Why or why not?

    No, $f(x)$ is undefined at $x = 0$. Division by zero makes the function undefined at that point.

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

    $7$. Direct substitution works for polynomial functions.

26. If $f(x)$ is continuous on $[0, 1]$ and $f(0) = 3$, can $f(1) = 0$?

    Yes, if $f(x)$ decreases continuously. IVT guarantees all values between $f(0)$ and $f(1)$ are achieved.

27. What is the definition of a jump discontinuity?

    When $\lim_{x \to a^-} f(x) \neq \lim_{x \to a^+} f(x)$. Left and right approaches yield different limit values.

28. State the condition for $f(x)$ to be continuous at an endpoint $a$ of $[a, b]$.

    $\text{lim}_{x \to a^+} f(x) = f(a)$. Only right-hand continuity needed at left endpoint.

29. Determine if $f(x) = \frac{x^2 - 9}{x - 3}$ is continuous at $x = 3$.

    No, $f(x)$ has a removable discontinuity. Factor: $\frac{(x+3)(x-3)}{x-3} = x+3$, but $f(3)$ undefined.

30. Identify the discontinuity: $\text{lim}_{x \to a} f(x)$ exists but $f(a)$ is not defined.

    Removable discontinuity. The 'hole' can be filled by defining the function at that point.

31. What is the condition for a function to be continuous over an open interval $(a, b)$?

    $f(x)$ is continuous at every point in $(a, b)$. No endpoint conditions needed for open intervals.

32. Find the value of $c$ that makes $f(x) = cx + 1$ continuous at $x = 2$ if $f(2) = 5$.

    $c = 2$. Set $2c + 1 = 5$ and solve: $c = 2$.

33. State the Intermediate Value Theorem.

    If $f(x)$ is continuous on $[a, b]$, $f(a) < N < f(b)$ then $\text{exists } c \text{ such that } f(c) = N$. Guarantees all intermediate values exist for continuous functions on closed intervals.

34. What is the condition for continuity from the left at $x = a$?

    $\text{lim}_{x \to a^-} f(x) = f(a)$. Left-hand limit must equal the function value.

35. For $f(x) = \frac{x^3 - 8}{x - 2}$, does $f(x)$ have a removable discontinuity at $x = 2$?

    Yes, $f(x)$ has a removable discontinuity. Factor: $\frac{(x-2)(x^2+2x+4)}{x-2} = x^2+2x+4$, limit exists.

36. Identify the type of discontinuity if $\text{lim}_{x \to a} f(x)$ does not exist.

    This is an infinite or jump discontinuity. When limits don't exist, the function has a break or approaches infinity.

Study these flashcards