Exploring Types of Discontinuities - AP Calculus AB
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Identify the discontinuity in $f(x) = \frac{1}{x^2}$ at $x = 0$.
Identify the discontinuity in $f(x) = \frac{1}{x^2}$ at $x = 0$.
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Infinite discontinuity. Positive denominator creates vertical asymptote as function approaches $+\infty$.
Infinite discontinuity. Positive denominator creates vertical asymptote as function approaches $+\infty$.
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What is the difference between removable and non-removable discontinuities?
What is the difference between removable and non-removable discontinuities?
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Removable can be fixed by redefining; non-removable cannot. Removable means "fixable"; non-removable means permanent breaks or asymptotes.
Removable can be fixed by redefining; non-removable cannot. Removable means "fixable"; non-removable means permanent breaks or asymptotes.
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Define a point of discontinuity.
Define a point of discontinuity.
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A point where a function is not continuous. Any location where the function fails to meet continuity requirements.
A point where a function is not continuous. Any location where the function fails to meet continuity requirements.
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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 left and right limits creating a "jump" in the graph.
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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What is a removable discontinuity?
What is a removable discontinuity?
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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.
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.
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What is the condition for a function to have a removable discontinuity at $x = c$?
What is the condition for a function to have a removable discontinuity at $x = c$?
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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.
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.
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Find the discontinuity type for $g(x) = \frac{x+2}{x^2-4}$ at $x = -2$.
Find the discontinuity type for $g(x) = \frac{x+2}{x^2-4}$ at $x = -2$.
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Removable discontinuity. Factor: $\frac{x+2}{(x+2)(x-2)}$, cancels to $\frac{1}{x-2}$ with hole at $x=-2$.
Removable discontinuity. Factor: $\frac{x+2}{(x+2)(x-2)}$, cancels to $\frac{1}{x-2}$ with hole at $x=-2$.
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Identify the discontinuity: $f(x) = \tan(x)$ at $x = \frac{\pi}{2}$.
Identify the discontinuity: $f(x) = \tan(x)$ 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 condition creates an infinite discontinuity?
What condition creates an infinite discontinuity?
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The function approaches infinity or negative infinity at the point. Denominator approaches zero while numerator approaches nonzero value.
The function approaches infinity or negative infinity at the point. Denominator approaches zero while numerator approaches nonzero value.
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Identify the type of discontinuity in $f(x) = \frac{x^2 - 1}{x - 1}$ at $x = 1$.
Identify the type of discontinuity in $f(x) = \frac{x^2 - 1}{x - 1}$ at $x = 1$.
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Removable discontinuity. Factor: $\frac{(x+1)(x-1)}{x-1} = x+1$, undefined at $x=1$ but limit exists.
Removable discontinuity. Factor: $\frac{(x+1)(x-1)}{x-1} = x+1$, undefined at $x=1$ but limit exists.
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What is the discontinuity in $f(x) = \frac{x^2+2x+1}{x+1}$ at $x = -1$?
What is the discontinuity in $f(x) = \frac{x^2+2x+1}{x+1}$ at $x = -1$?
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Removable discontinuity. Factor: $\frac{(x+1)^2}{x+1}$, simplifies to $(x+1)$ with hole at $x=-1$.
Removable discontinuity. Factor: $\frac{(x+1)^2}{x+1}$, simplifies to $(x+1)$ with hole at $x=-1$.
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Identify the discontinuity in $g(x) = \frac{1}{(x-2)^2}$ at $x = 2$.
Identify the discontinuity in $g(x) = \frac{1}{(x-2)^2}$ at $x = 2$.
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Infinite discontinuity. Squared denominator ensures function approaches $+\infty$ from both sides.
Infinite discontinuity. Squared denominator ensures function approaches $+\infty$ from both sides.
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What type of discontinuity does $f(x) = \lfloor x \rfloor$ have at integer points?
What type of discontinuity does $f(x) = \lfloor x \rfloor$ have at integer points?
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Jump discontinuity. Floor function creates jumps of size 1 at every integer value.
Jump discontinuity. Floor function creates jumps of size 1 at every integer value.
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Identify the discontinuity in $f(x) = \frac{1}{x}$ at $x = 0$.
Identify the discontinuity in $f(x) = \frac{1}{x}$ at $x = 0$.
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Infinite discontinuity. Division by zero creates a vertical asymptote as $f(x) \to \pm\infty$.
Infinite discontinuity. Division by zero creates a vertical asymptote as $f(x) \to \pm\infty$.
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Identify the discontinuity for $f(x) = \frac{x}{x^2 - 4}$ at $x = 2$.
Identify the discontinuity for $f(x) = \frac{x}{x^2 - 4}$ at $x = 2$.
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Infinite discontinuity. Factor denominator: $x^2-4=(x+2)(x-2)$, zero at $x=2$.
Infinite discontinuity. Factor denominator: $x^2-4=(x+2)(x-2)$, zero at $x=2$.
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What type of discontinuity is present in $f(x) = \frac{x^2 - 9}{x-3}$ at $x = 3$?
What type of discontinuity is present in $f(x) = \frac{x^2 - 9}{x-3}$ at $x = 3$?
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Removable discontinuity. Factor: $\frac{(x+3)(x-3)}{x-3}$, creates removable hole at $x=3$.
Removable discontinuity. Factor: $\frac{(x+3)(x-3)}{x-3}$, creates removable hole at $x=3$.
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Determine if $h(x) = |x|$ has any discontinuities.
Determine if $h(x) = |x|$ has any discontinuities.
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No discontinuities. Absolute value function is continuous everywhere with no breaks or jumps.
No discontinuities. Absolute value function is continuous everywhere with no breaks or jumps.
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Identify the discontinuity in $f(x) = \frac{x^2 - 1}{x+1}$ at $x = -1$.
Identify the discontinuity in $f(x) = \frac{x^2 - 1}{x+1}$ at $x = -1$.
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Removable discontinuity. Factor: $\frac{(x+1)(x-1)}{x+1}$, creates removable hole at $x=-1$.
Removable discontinuity. Factor: $\frac{(x+1)(x-1)}{x+1}$, creates removable hole at $x=-1$.
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Determine the discontinuity for $f(x) = \frac{x^2 + 4x + 4}{x + 2}$ at $x = -2$.
Determine the discontinuity for $f(x) = \frac{x^2 + 4x + 4}{x + 2}$ at $x = -2$.
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Removable discontinuity. Factor: $\frac{(x+2)^2}{x+2}$, simplifies to $(x+2)$ with hole at $x=-2$.
Removable discontinuity. Factor: $\frac{(x+2)^2}{x+2}$, simplifies to $(x+2)$ with hole at $x=-2$.
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What is an infinite discontinuity?
What is an infinite discontinuity?
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A discontinuity where a function approaches infinity at a point. The function has a vertical asymptote where values grow without bound.
A discontinuity where a function approaches infinity at a point. The function has a vertical asymptote where values grow without bound.
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Determine discontinuity type in $f(x) = \frac{1}{x^2 - 1}$ at $x = 1$.
Determine discontinuity type in $f(x) = \frac{1}{x^2 - 1}$ at $x = 1$.
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Infinite discontinuity. Factor $x^2-1=(x+1)(x-1)$, so denominator is zero at $x=1$.
Infinite discontinuity. Factor $x^2-1=(x+1)(x-1)$, so denominator is zero at $x=1$.
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What is the definition of a function being discontinuous at a point?
What is the definition of a function being discontinuous at a point?
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The function is not continuous at that point. The function violates at least one of the three continuity conditions.
The function is not continuous at that point. The function violates at least one of the three continuity conditions.
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What type of discontinuity is in $f(x) = \frac{1}{x(x-3)}$ at $x = 3$?
What type of discontinuity is in $f(x) = \frac{1}{x(x-3)}$ at $x = 3$?
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Infinite discontinuity. Denominator has factors $x$ and $(x-3)$, both creating vertical asymptotes.
Infinite discontinuity. Denominator has factors $x$ and $(x-3)$, both creating vertical asymptotes.
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Which type of discontinuity occurs when $\lim_{x \to c^-} f(x) \neq \lim_{x \to c^+} f(x)$?
Which type of discontinuity occurs when $\lim_{x \to c^-} f(x) \neq \lim_{x \to c^+} f(x)$?
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Jump discontinuity. When one-sided limits exist but are unequal, the function "jumps" between values.
Jump discontinuity. When one-sided limits exist but are unequal, the function "jumps" between values.
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Identify the discontinuity in $f(x) = \lfloor x \rfloor$ at $x = 1$.
Identify the discontinuity in $f(x) = \lfloor x \rfloor$ at $x = 1$.
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Jump discontinuity. Left limit is 0, right limit is 1, creating a jump of size 1.
Jump discontinuity. Left limit is 0, right limit is 1, creating a jump of size 1.
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What condition indicates a function is discontinuous at a point?
What condition indicates a function is discontinuous at a point?
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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.
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.
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What is an essential discontinuity?
What is an essential discontinuity?
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A discontinuity that is neither removable nor a jump. Includes oscillatory discontinuities where limits don't exist in any form.
A discontinuity that is neither removable nor a jump. Includes oscillatory discontinuities where limits don't exist in any form.
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What type of discontinuity is present in $f(x) = \frac{x^2 - x - 2}{x - 2}$ at $x = 2$?
What type of discontinuity is present in $f(x) = \frac{x^2 - x - 2}{x - 2}$ at $x = 2$?
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Removable discontinuity. Factor: $\frac{(x-2)(x+1)}{x-2}$, creates hole at $x=2$.
Removable discontinuity. Factor: $\frac{(x-2)(x+1)}{x-2}$, creates hole at $x=2$.
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What type of discontinuity occurs if $\lim_{x \to c} f(x)$ does not exist?
What type of discontinuity occurs if $\lim_{x \to c} f(x)$ does not exist?
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Non-removable discontinuity. When the limit fails to exist, the discontinuity cannot be removed.
Non-removable discontinuity. When the limit fails to exist, the discontinuity cannot be removed.
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Determine the discontinuity for $f(x) = \frac{1}{x-3}$ at $x = 3$.
Determine the discontinuity for $f(x) = \frac{1}{x-3}$ at $x = 3$.
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Infinite discontinuity. Denominator equals zero creating vertical asymptote where function approaches infinity.
Infinite discontinuity. Denominator equals zero creating vertical asymptote where function approaches infinity.
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