Exploring Types of Discontinuities - AP Calculus BC
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What is a jump discontinuity?
What is a jump discontinuity?
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A discontinuity where the left and right limits exist but are not equal. The function has different values when approached from left vs right.
A discontinuity where the left and right limits exist but are not equal. The function has different values when approached from left vs right.
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State the definition of an infinite discontinuity.
State the definition of an infinite discontinuity.
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A discontinuity where the function approaches infinity at a point. The function grows without bound, creating a vertical asymptote.
A discontinuity where the function approaches infinity at a point. The function grows without bound, creating a vertical asymptote.
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Identify the discontinuity type for $f(x) = \frac{1}{x}$ at $x = 0$.
Identify the discontinuity type for $f(x) = \frac{1}{x}$ at $x = 0$.
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Infinite discontinuity. Division by zero causes the function to approach $\pm\infty$.
Infinite discontinuity. Division by zero causes the function to approach $\pm\infty$.
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What type of discontinuity is characterized by a vertical asymptote?
What type of discontinuity is characterized by a vertical asymptote?
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Infinite discontinuity. The function approaches infinity, creating a vertical line.
Infinite discontinuity. The function approaches infinity, creating a vertical line.
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What is the primary characteristic of a removable discontinuity?
What is the primary characteristic of a removable discontinuity?
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The limit exists but the function is not defined at that point. A hole exists that can be filled by defining the limit value.
The limit exists but the function is not defined at that point. A hole exists that can be filled by defining the limit value.
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What type of discontinuity occurs if both one-sided limits exist and are finite but unequal?
What type of discontinuity occurs if both one-sided limits exist and are finite but unequal?
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Jump discontinuity. Left and right limits exist but have different finite values.
Jump discontinuity. Left and right limits exist but have different finite values.
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Find the discontinuity for $f(x) = \frac{x^2 - 4}{x - 2}$ at $x=2$.
Find the discontinuity for $f(x) = \frac{x^2 - 4}{x - 2}$ at $x=2$.
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Removable discontinuity. Factor $(x-2)$ cancels, creating a hole at $x = 2$.
Removable discontinuity. Factor $(x-2)$ cancels, creating a hole at $x = 2$.
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Which type of discontinuity can be removed by redefining the function at a point?
Which type of discontinuity can be removed by redefining the function at a point?
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Removable discontinuity. Simply define the function value at the point to make it continuous.
Removable discontinuity. Simply define the function value at the point to make it continuous.
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What type of discontinuity occurs when a limit does not exist at a point?
What type of discontinuity occurs when a limit does not exist at a point?
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Infinite or jump discontinuity. No limit means either infinite behavior or unequal one-sided limits.
Infinite or jump discontinuity. No limit means either infinite behavior or unequal one-sided limits.
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What is a key feature of a function with a jump discontinuity?
What is a key feature of a function with a jump discontinuity?
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Unequal one-sided limits. Left and right approaches yield different finite values.
Unequal one-sided limits. Left and right approaches yield different finite values.
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Identify the type of discontinuity for $f(x) = \frac{1}{x(x-2)}$ at $x = 0$.
Identify the type of discontinuity for $f(x) = \frac{1}{x(x-2)}$ at $x = 0$.
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Infinite discontinuity. Denominator zero makes function approach infinity at $x = 0$.
Infinite discontinuity. Denominator zero makes function approach infinity at $x = 0$.
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What type of discontinuity occurs when the function is only undefined at a point?
What type of discontinuity occurs when the function is only undefined at a point?
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Removable discontinuity. The limit exists but the function value doesn't at that point.
Removable discontinuity. The limit exists but the function value doesn't at that point.
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Identify the discontinuity type for $f(x) = \ln(x)$ at $x = 0$.
Identify the discontinuity type for $f(x) = \ln(x)$ at $x = 0$.
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Infinite discontinuity. Natural log approaches $-\infty$ as $x$ approaches zero from right.
Infinite discontinuity. Natural log approaches $-\infty$ as $x$ approaches zero from right.
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Identify the type of discontinuity at $x = a$ for $f(x) = \frac{x^2 - a^2}{x - a}$.
Identify the type of discontinuity at $x = a$ for $f(x) = \frac{x^2 - a^2}{x - a}$.
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Removable discontinuity. The factor $ (x-a) $ cancels, leaving a hole at $x = a$.
Removable discontinuity. The factor $ (x-a) $ cancels, leaving a hole at $x = a$.
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Find the discontinuity type for $f(x) = \tan(x)$ at $x = \frac{\pi}{2}$.
Find the discontinuity type for $f(x) = \tan(x)$ at $x = \frac{\pi}{2}$.
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Infinite discontinuity. Tangent has vertical asymptotes where cosine equals zero.
Infinite discontinuity. Tangent has vertical asymptotes where cosine equals zero.
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What is a removable discontinuity in a function?
What is a removable discontinuity in a function?
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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.
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.
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What type of discontinuity is present in a piecewise function with a gap?
What type of discontinuity is present in a piecewise function with a gap?
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Jump discontinuity. Different function definitions create unequal one-sided limits.
Jump discontinuity. Different function definitions create unequal one-sided limits.
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What is the effect of redefining a function at a removable discontinuity?
What is the effect of redefining a function at a removable discontinuity?
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It can make the function continuous at that point. Filling the hole makes the function continuous at that point.
It can make the function continuous at that point. Filling the hole makes the function continuous at that point.
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Identify the discontinuity type for $f(x) = \frac{1}{x^2}$ at $x = 0$.
Identify the discontinuity type for $f(x) = \frac{1}{x^2}$ at $x = 0$.
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Infinite discontinuity. Function approaches $+\infty$ from both sides at $x = 0$.
Infinite discontinuity. Function approaches $+\infty$ from both sides at $x = 0$.
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Find the type of discontinuity for $f(x) = \frac{x^3 - 27}{x - 3}$ at $x = 3$.
Find the type of discontinuity for $f(x) = \frac{x^3 - 27}{x - 3}$ at $x = 3$.
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Removable discontinuity. Factor $ (x-3) $ cancels from numerator and denominator.
Removable discontinuity. Factor $ (x-3) $ cancels from numerator and denominator.
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Identify the discontinuity type for $f(x) = \frac{1}{(x-3)^2}$ at $x = 3$.
Identify the discontinuity type for $f(x) = \frac{1}{(x-3)^2}$ at $x = 3$.
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Infinite discontinuity. Denominator approaches zero while numerator doesn't at $x = 3$.
Infinite discontinuity. Denominator approaches zero while numerator doesn't at $x = 3$.
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What type of discontinuity is resolved by redefining the function's value?
What type of discontinuity is resolved by redefining the function's value?
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Removable discontinuity. Assigning the limit value makes the function continuous.
Removable discontinuity. Assigning the limit value makes the function continuous.
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Identify the discontinuity type for $f(x) = \frac{|x|}{x}$ at $x = 0$.
Identify the discontinuity type for $f(x) = \frac{|x|}{x}$ at $x = 0$.
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Jump discontinuity. Left limit $-1$, right limit $1$, function undefined at origin.
Jump discontinuity. Left limit $-1$, right limit $1$, function undefined at origin.
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Find the type of discontinuity for $f(x) = \frac{x^2 - 1}{x - 1}$ at $x = 1$.
Find the type of discontinuity for $f(x) = \frac{x^2 - 1}{x - 1}$ at $x = 1$.
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Removable discontinuity. Factor $(x-1)$ cancels, giving limit value of $2$.
Removable discontinuity. Factor $(x-1)$ cancels, giving limit value of $2$.
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Find the type of discontinuity for $f(x) = \frac{1}{(x-1)(x+1)}$ at $x = 1$.
Find the type of discontinuity for $f(x) = \frac{1}{(x-1)(x+1)}$ at $x = 1$.
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Infinite discontinuity. Denominator becomes zero while numerator doesn't at $x = 1$.
Infinite discontinuity. Denominator becomes zero while numerator doesn't at $x = 1$.
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What type of discontinuity is present if a function has a vertical asymptote?
What type of discontinuity is present if a function has a vertical asymptote?
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Infinite discontinuity. The graph has a vertical line where function approaches infinity.
Infinite discontinuity. The graph has a vertical line where function approaches infinity.
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Identify the type of discontinuity for $f(x) = |x|/x$ at $x = 0$.
Identify the type of discontinuity for $f(x) = |x|/x$ at $x = 0$.
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Jump discontinuity. Left limit is $-1$, right limit is $1$, but function undefined.
Jump discontinuity. Left limit is $-1$, right limit is $1$, but function undefined.
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What type of discontinuity is present when $f(x)$ is defined piecewise with unequal limits?
What type of discontinuity is present when $f(x)$ is defined piecewise with unequal limits?
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Jump discontinuity. Different function values on either side create a gap in the graph.
Jump discontinuity. Different function values on either side create a gap in the graph.
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Find the type of discontinuity for $f(x) = \frac{x^2 - 16}{x - 4}$ at $x = 4$.
Find the type of discontinuity for $f(x) = \frac{x^2 - 16}{x - 4}$ at $x = 4$.
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Removable discontinuity. Factor $(x-4)$ cancels, creating a hole at $x = 4$.
Removable discontinuity. Factor $(x-4)$ cancels, creating a hole at $x = 4$.
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What type of discontinuity is associated with a rational function's hole?
What type of discontinuity is associated with a rational function's hole?
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Removable discontinuity. A hole in the graph that can be filled.
Removable discontinuity. A hole in the graph that can be filled.
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