AP Calculus BC Flashcards: Exploring Types Of Discontinuities

What this deck covers

This deck focuses on Exploring Types Of Discontinuities, giving you a quick way to review the definitions, rules, and examples that matter most for AP Calculus BC.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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

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

Flashcard 2: Identify the discontinuity type for $f(x) = \ln(x)$ at $x = 0$.

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

Flashcard 3: What happens to the graph at a jump discontinuity?

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

Flashcard 4: What is the primary characteristic of a removable discontinuity?

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

Flashcard 5: What is a removable discontinuity in a function?

Answer: 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.

Flashcard 6: What type of discontinuity is characterized by a vertical asymptote?

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

Flashcard 7: Identify the discontinuity type for $f(x) = \frac{1}{x^3}$ at $x = 0$.

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

Flashcard 8: Find the type of discontinuity for $f(x) = \frac{x^2 - 16}{x - 4}$ at $x = 4$.

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

Flashcard 9: What type of discontinuity is associated with a rational function's hole?

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

Flashcard 10: Identify the discontinuity type for $f(x) = \frac{1}{x^2}$ at $x = 0$.

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

Flashcard 11: What is the effect of redefining a function at a removable discontinuity?

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

Flashcard 12: What type of discontinuity occurs if both one-sided limits exist and are finite but unequal?

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

Flashcard 13: What type of discontinuity is characterized by a vertical asymptote?

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

Flashcard 14: Identify the type of discontinuity for $f(x) = |x|/x$ at $x = 0$.

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

Flashcard 15: Identify the discontinuity type for $f(x) = \frac{1}{(x-3)^2}$ at $x = 3$.

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

Flashcard 16: Identify the discontinuity type for $f(x) = \frac{|x|}{x}$ at $x = 0$.

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

Flashcard 17: What type of discontinuity is present in a piecewise function with a gap?

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

Flashcard 18: What type of discontinuity is present in a piecewise function with a gap?

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

Flashcard 19: Identify the discontinuity type for $f(x) = \frac{|x|}{x}$ at $x = 0$.

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

Flashcard 20: Identify the discontinuity type for $f(x) = \frac{1}{(x-3)^2}$ at $x = 3$.

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

Flashcard 21: Find the type of discontinuity for $f(x) = \frac{x^3 - 27}{x - 3}$ at $x = 3$.

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

Flashcard 22: Identify the discontinuity type for $f(x) = \ln(x)$ at $x = 0$.

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

Flashcard 23: What is the primary characteristic of a removable discontinuity?

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

Flashcard 24: Identify the discontinuity type for $f(x) = \frac{1}{x}$ at $x = 0$.

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

Flashcard 25: What type of discontinuity occurs when the function is only undefined at a point?

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

Flashcard 26: Find the type of discontinuity for $f(x) = \frac{1}{(x-1)(x+1)}$ at $x = 1$.

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

Flashcard 27: Identify the discontinuity type for $f(x) = \frac{1}{x^3}$ at $x = 0$.

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

Flashcard 28: What type of discontinuity is often resolved by canceling common factors?

Answer: Removable discontinuity. Canceling common factors reveals the removable nature.

Flashcard 29: What type of discontinuity is present for $f(x) = \frac{1}{x-1}$ at $x = 1$?

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

Flashcard 30: What is a key feature of a function with a jump discontinuity?

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

Flashcard 31: What type of discontinuity does $f(x) = [x]$ have at integer values?

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

Flashcard 32: What type of discontinuity is present when $f(x)$ is defined piecewise with unequal limits?

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

Flashcard 33: What type of discontinuity is often resolved by canceling common factors?

Answer: Removable discontinuity. Canceling common factors reveals the removable nature.

Flashcard 34: Find the type of discontinuity for $f(x) = \frac{x^2 - 1}{x - 1}$ at $x = 1$.

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

Flashcard 35: Which type of discontinuity can be removed by redefining the function at a point?

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

Flashcard 36: What is a removable discontinuity in a function?

Answer: 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.

Flashcard 37: Find the discontinuity type for $f(x) = \frac{x^2 - 9}{x - 3}$ at $x = 3$.

Answer: Removable discontinuity. Factor $(x-3)$ cancels, leaving $(x+3) = 6$ at $x = 3$.

Flashcard 38: What type of discontinuity is present if a function has a vertical asymptote?

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

Flashcard 39: What type of discontinuity is resolved by redefining the function's value?

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

Flashcard 40: What type of discontinuity occurs when a limit does not exist at a point?

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

Flashcard 41: Which type of discontinuity can be removed by redefining the function at a point?

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

Flashcard 42: What type of discontinuity is present when $f(x)$ is defined piecewise with unequal limits?

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

Flashcard 43: Identify the type of discontinuity at $x = a$ for $f(x) = \frac{x^2 - a^2}{x - a}$.

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

Flashcard 44: What type of discontinuity occurs when the function is only undefined at a point?

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

Flashcard 45: State the definition of an infinite discontinuity.

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

Flashcard 46: Identify the type of discontinuity for $f(x) = \frac{1}{x(x-2)}$ at $x = 0$.

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

Flashcard 47: Find the type of discontinuity for $f(x) = \frac{x^2 - 16}{x - 4}$ at $x = 4$.

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

Flashcard 48: What type of discontinuity is associated with a rational function's hole?

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

Flashcard 49: Identify the type of discontinuity at $x = a$ for $f(x) = \frac{x^2 - a^2}{x - a}$.

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

Flashcard 50: Find the type of discontinuity for $f(x) = \frac{x^2 - 1}{x - 1}$ at $x = 1$.

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

Flashcard 51: What type of discontinuity occurs if both one-sided limits exist and are finite but unequal?

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

Flashcard 52: Identify the discontinuity type for $f(x) = \frac{1}{x^2}$ at $x = 0$.

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

Flashcard 53: Find the discontinuity for $f(x) = \frac{x^2 - 4}{x - 2}$ at $x=2$.

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

Flashcard 54: Find the type of discontinuity for $f(x) = \frac{x^2 - 25}{x - 5}$ at $x = 5$.

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

Flashcard 55: What is a key feature of a function with a jump discontinuity?

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

Flashcard 56: What type of discontinuity does $f(x) = [x]$ have at integer values?

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

Flashcard 57: What is a jump discontinuity?

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

Flashcard 58: Find the discontinuity for $f(x) = \frac{x^2 - 4}{x - 2}$ at $x=2$.

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

Flashcard 59: What is a jump discontinuity?

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

Flashcard 60: What happens to the graph at a jump discontinuity?

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

Flashcard 61: Find the type of discontinuity for $f(x) = \frac{x^2 - 25}{x - 5}$ at $x = 5$.

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

Flashcard 62: Find the discontinuity type for $f(x) = \tan(x)$ at $x = \frac{\pi}{2}$.

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

Flashcard 63: What is the effect of redefining a function at a removable discontinuity?

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

Flashcard 64: Identify the type of discontinuity for $f(x) = \frac{1}{x(x-2)}$ at $x = 0$.

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

Flashcard 65: Find the type of discontinuity for $f(x) = \frac{1}{(x-1)(x+1)}$ at $x = 1$.

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

Flashcard 66: What type of discontinuity is resolved by redefining the function's value?

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

Flashcard 67: What type of discontinuity occurs when a limit does not exist at a point?

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

Flashcard 68: Find the discontinuity type for $f(x) = \tan(x)$ at $x = \frac{\pi}{2}$.

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

Flashcard 69: Find the discontinuity type for $f(x) = \frac{x^2 - 9}{x - 3}$ at $x = 3$.

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

Flashcard 70: Identify the discontinuity type for $f(x) = \frac{1}{x}$ at $x = 0$.

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

Flashcard 71: Find the type of discontinuity for $f(x) = \frac{x^3 - 27}{x - 3}$ at $x = 3$.

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

Flashcard 72: Identify the type of discontinuity for $f(x) = |x|/x$ at $x = 0$.

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

Flashcard 73: What type of discontinuity is present if a function has a vertical asymptote?

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

Flashcard 74: State the definition of an infinite discontinuity.

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