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 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 2: 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 3: 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 4: What type of discontinuity is characterized by a vertical asymptote?

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

Flashcard 5: 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 6: 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 7: 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 8: 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 9: 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 10: 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 11: 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 12: 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 13: 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 14: 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 15: 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 16: 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 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 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 19: 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 20: 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 21: 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 22: What type of discontinuity is resolved by redefining the function's value?

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

Flashcard 23: 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 24: 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 25: 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 26: 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 27: 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 28: 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 29: 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 30: What type of discontinuity is associated with a rational function's hole?

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