Exploring Types of Discontinuities - AP Calculus AB

1. Identify the discontinuity: $f(x) = \tan(x)$ at $x = \frac{\pi}{2}$.

   Infinite discontinuity. Tangent has vertical asymptotes where cosine equals zero.

2. Identify the discontinuity in $f(x) = \frac{x^2 - 1}{x+1}$ at $x = -1$.

   Removable discontinuity. Factor: $\frac{(x+1)(x-1)}{x+1}$, creates removable hole at $x=-1$.

3. Determine the discontinuity for $f(x) = \frac{x^2 + 4x + 4}{x + 2}$ at $x = -2$.

   Removable discontinuity. Factor: $\frac{(x+2)^2}{x+2}$, simplifies to $(x+2)$ with hole at $x=-2$.

4. What condition indicates a function is discontinuous at a point?

   The limit does not equal the function value or does not exist. Either the limit doesn't exist or doesn't equal the function value.

5. Identify the type of discontinuity in $f(x) = \frac{x^2 - 1}{x - 1}$ at $x = 1$.

   Removable discontinuity. Factor: $\frac{(x+1)(x-1)}{x-1} = x+1$, undefined at $x=1$ but limit exists.

6. Which type of discontinuity occurs when $\lim_{x \to c^-} f(x) \neq \lim_{x \to c^+} f(x)$?

   Jump discontinuity. When one-sided limits exist but are unequal, the function "jumps" between values.

7. What type of discontinuity does $f(x) = \lfloor x \rfloor$ have at integer points?

   Jump discontinuity. Floor function creates jumps of size 1 at every integer value.

8. Define a point of discontinuity.

   A point where a function is not continuous. Any location where the function fails to meet continuity requirements.

9. Determine discontinuity type in $f(x) = \frac{x^2 - 9}{x - 3}$ at $x = 3$.

   Removable discontinuity. Factor: $\frac{(x+3)(x-3)}{x-3}$, creates removable hole at $x=3$.

10. What type of discontinuity is present in $f(x) = \frac{x^2 - x - 2}{x - 2}$ at $x = 2$?

    Removable discontinuity. Factor: $\frac{(x-2)(x+1)}{x-2}$, creates hole at $x=2$.

11. What is the definition of a function being discontinuous at a point?

    The function is not continuous at that point. The function violates at least one of the three continuity conditions.

12. Identify the discontinuity in $f(x) = \lfloor x \rfloor$ at $x = 1$.

    Jump discontinuity. Left limit is 0, right limit is 1, creating a jump of size 1.

13. Find the discontinuity type for $g(x) = \frac{x+2}{x^2-4}$ at $x = -2$.

    Removable discontinuity. Factor: $\frac{x+2}{(x+2)(x-2)}$, cancels to $\frac{1}{x-2}$ with hole at $x=-2$.

14. What is the difference between removable and non-removable discontinuities?

    Removable can be fixed by redefining; non-removable cannot. Removable means "fixable"; non-removable means permanent breaks or asymptotes.

15. Identify the discontinuity for $f(x) = \frac{x}{x^2 - 4}$ at $x = 2$.

    Infinite discontinuity. Factor denominator: $x^2-4=(x+2)(x-2)$, zero at $x=2$.

16. Identify the discontinuity in $f(x) = \frac{1}{x^2}$ at $x = 0$.

    Infinite discontinuity. Positive denominator creates vertical asymptote as function approaches $+\infty$.

17. What type of discontinuity occurs if $\lim_{x \to c} f(x)$ does not exist?

    Non-removable discontinuity. When the limit fails to exist, the discontinuity cannot be removed.

18. What type of discontinuity does $f(x) = \frac{1}{x(x-1)}$ have at $x = 0$?

    Infinite discontinuity. Both factors in denominator create vertical asymptotes at $x=0$ and $x=1$.

19. What is an infinite discontinuity?

    A discontinuity where a function approaches infinity at a point. The function has a vertical asymptote where values grow without bound.

20. What does it mean if a function is continuous at a point?

    The function's limit, value, and limit from both sides are equal. All three continuity conditions: function exists, limit exists, and they're equal.

21. Identify the discontinuity in $g(x) = \frac{1}{(x-2)^2}$ at $x = 2$.

    Infinite discontinuity. Squared denominator ensures function approaches $+\infty$ from both sides.

22. What is the condition for a function to have a removable discontinuity at $x = c$?

    The limit exists, but the function value is either undefined or different. The limit approaches a finite value, but function is undefined or has wrong value.

23. Determine the discontinuity for $f(x) = \frac{1}{x-3}$ at $x = 3$.

    Infinite discontinuity. Denominator equals zero creating vertical asymptote where function approaches infinity.

24. What is a removable discontinuity?

    A discontinuity that can be removed by redefining the function. The function has a hole that can be filled by defining the value at that point.

25. What condition creates an infinite discontinuity?

    The function approaches infinity or negative infinity at the point. Denominator approaches zero while numerator approaches nonzero value.

26. Determine if $h(x) = |x|$ has any discontinuities.

    No discontinuities. Absolute value function is continuous everywhere with no breaks or jumps.

27. Determine discontinuity type in $f(x) = \frac{1}{x^2 - 1}$ at $x = 1$.

    Infinite discontinuity. Factor $x^2-1=(x+1)(x-1)$, so denominator is zero at $x=1$.

28. Identify the discontinuity in $f(x) = \frac{x+3}{x^2-9}$ at $x = -3$.

    Removable discontinuity. Factor: $\frac{x+3}{(x+3)(x-3)}$, simplifies with hole at $x=-3$.

29. Identify the discontinuity: $f(x) = \frac{x^2 - 4}{x - 2}$ at $x = 2$.

    Removable discontinuity. Factor: $\frac{(x+2)(x-2)}{x-2}$, simplifies with removable hole at $x=2$.

30. What condition creates a jump discontinuity at $x = c$?

    The left-hand and right-hand limits at $c$ exist but differ. The graph has a finite "jump" between the left and right limit values.

31. What is an essential discontinuity?

    A discontinuity that is neither removable nor a jump. Includes oscillatory discontinuities where limits don't exist in any form.

32. What type of discontinuity is in $f(x) = \frac{1}{x(x-3)}$ at $x = 3$?

    Infinite discontinuity. Denominator has factors $x$ and $(x-3)$, both creating vertical asymptotes.

33. What is the definition of continuity on an interval?

    A function is continuous if it is continuous at every point on the interval. No breaks, holes, or jumps exist anywhere within the given interval.

34. Identify the discontinuity in $f(x) = \frac{1}{x}$ at $x = 0$.

    Infinite discontinuity. Division by zero creates a vertical asymptote as $f(x) \to \pm\infty$.

35. What is the discontinuity in $f(x) = \frac{x^2+2x+1}{x+1}$ at $x = -1$?

    Removable discontinuity. Factor: $\frac{(x+1)^2}{x+1}$, simplifies to $(x+1)$ with hole at $x=-1$.

36. What type of discontinuity is present in $f(x) = \frac{x^2 - 9}{x-3}$ at $x = 3$?

    Removable discontinuity. Factor: $\frac{(x+3)(x-3)}{x-3}$, creates removable hole at $x=3$.

37. What is required for a function to be continuous at a point?

    The limit and function value must be equal at that point. The function value must exist and equal the limit at that point.

38. What is a jump discontinuity?

    A discontinuity where the left and right limits exist but are not equal. The function has different left and right limits creating a "jump" in the graph.

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