Exploring Types of Discontinuities - AP Calculus BC

1. Identify the discontinuity type for $f(x) = \ln(x)$ at $x = 0$.

   Infinite discontinuity. Natural log approaches $-\infty$ as $x$ approaches zero from right.

2. What is the primary characteristic of a removable discontinuity?

   The limit exists but the function is not defined at that point. A hole exists that can be filled by defining the limit value.

3. Find the type of discontinuity for $f(x) = \frac{x^2 - 16}{x - 4}$ at $x = 4$.

   Removable discontinuity. Factor $(x-4)$ cancels, creating a hole at $x = 4$.

4. What type of discontinuity is associated with a rational function's hole?

   Removable discontinuity. A hole in the graph that can be filled.

5. Identify the discontinuity type for $f(x) = \frac{1}{x^2}$ at $x = 0$.

   Infinite discontinuity. Function approaches $+\infty$ from both sides at $x = 0$.

6. What is the effect of redefining a function at a removable discontinuity?

   It can make the function continuous at that point. Filling the hole makes the function continuous at that point.

7. What type of discontinuity occurs if both one-sided limits exist and are finite but unequal?

   Jump discontinuity. Left and right limits exist but have different finite values.

8. What type of discontinuity is characterized by a vertical asymptote?

   Infinite discontinuity. The function approaches infinity, creating a vertical line.

9. Identify the type of discontinuity for $f(x) = |x|/x$ at $x = 0$.

   Jump discontinuity. Left limit is $-1$, right limit is $1$, but function undefined.

10. Identify the discontinuity type for $f(x) = \frac{1}{(x-3)^2}$ at $x = 3$.

    Infinite discontinuity. Denominator approaches zero while numerator doesn't at $x = 3$.

11. What type of discontinuity is present in a piecewise function with a gap?

    Jump discontinuity. Different function definitions create unequal one-sided limits.

12. Identify the discontinuity type for $f(x) = \frac{|x|}{x}$ at $x = 0$.

    Jump discontinuity. Left limit $-1$, right limit $1$, function undefined at origin.

13. Find the type of discontinuity for $f(x) = \frac{1}{(x-1)(x+1)}$ at $x = 1$.

    Infinite discontinuity. Denominator becomes zero while numerator doesn't at $x = 1$.

14. Identify the discontinuity type for $f(x) = \frac{1}{x^3}$ at $x = 0$.

    Infinite discontinuity. Function approaches $+\infty$ from both sides at $x = 0$.

15. What type of discontinuity is present for $f(x) = \frac{1}{x-1}$ at $x = 1$?

    Infinite discontinuity. Denominator becomes zero causing function to approach infinity.

16. What type of discontinuity is present when $f(x)$ is defined piecewise with unequal limits?

    Jump discontinuity. Different function values on either side create a gap in the graph.

17. What type of discontinuity is often resolved by canceling common factors?

    Removable discontinuity. Canceling common factors reveals the removable nature.

18. What is a removable discontinuity in a function?

    A point discontinuity where a limit exists but the function is undefined. The function has a hole that can be filled by defining the limit value.

19. What type of discontinuity is present if a function has a vertical asymptote?

    Infinite discontinuity. The graph has a vertical line where function approaches infinity.

20. What type of discontinuity is resolved by redefining the function's value?

    Removable discontinuity. Assigning the limit value makes the function continuous.

21. Which type of discontinuity can be removed by redefining the function at a point?

    Removable discontinuity. Simply define the function value at the point to make it continuous.

22. Identify the type of discontinuity at $x = a$ for $f(x) = \frac{x^2 - a^2}{x - a}$.

    Removable discontinuity. The factor $ (x-a) $ cancels, leaving a hole at $x = a$.

23. What type of discontinuity occurs when the function is only undefined at a point?

    Removable discontinuity. The limit exists but the function value doesn't at that point.

24. Find the type of discontinuity for $f(x) = \frac{x^2 - 1}{x - 1}$ at $x = 1$.

    Removable discontinuity. Factor $(x-1)$ cancels, giving limit value of $2$.

25. What is a key feature of a function with a jump discontinuity?

    Unequal one-sided limits. Left and right approaches yield different finite values.

26. What type of discontinuity does $f(x) = [x]$ have at integer values?

    Jump discontinuity. Floor function has different left and right limits at integers.

27. What is a jump discontinuity?

    A discontinuity where the left and right limits exist but are not equal. The function has different values when approached from left vs right.

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

    Removable discontinuity. Factor $(x-2)$ cancels, creating a hole at $x = 2$.

29. What happens to the graph at a jump discontinuity?

    The graph has a sudden break or jump at the point. The graph has a visible gap between different function levels.

30. Find the type of discontinuity for $f(x) = \frac{x^2 - 25}{x - 5}$ at $x = 5$.

    Removable discontinuity. Factor $ (x-5) $ cancels, leaving $ (x+5) = 10 $ at $x = 5$.

31. Find the discontinuity type for $f(x) = \tan(x)$ at $x = \frac{\pi}{2}$.

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

32. Identify the type of discontinuity for $f(x) = \frac{1}{x(x-2)}$ at $x = 0$.

    Infinite discontinuity. Denominator zero makes function approach infinity at $x = 0$.

33. What type of discontinuity occurs when a limit does not exist at a point?

    Infinite or jump discontinuity. No limit means either infinite behavior or unequal one-sided limits.

34. Find the discontinuity type for $f(x) = \frac{x^2 - 9}{x - 3}$ at $x = 3$.

    Removable discontinuity. Factor $x-3$ cancels, leaving $x+3 = 6$ at $x = 3$.

35. Identify the discontinuity type for $f(x) = \frac{1}{x}$ at $x = 0$.

    Infinite discontinuity. Division by zero causes the function to approach $\pm\infty$.

36. Find the type of discontinuity for $f(x) = \frac{x^3 - 27}{x - 3}$ at $x = 3$.

    Removable discontinuity. Factor $ (x-3) $ cancels from numerator and denominator.

37. State the definition of an infinite discontinuity.

    A discontinuity where the function approaches infinity at a point. The function grows without bound, creating a vertical asymptote.

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