Confirming Continuity over an Interval - AP Calculus AB
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Find the limit: $\text{lim}_{x \to 3} (2x + 1)$.
Find the limit: $\text{lim}_{x \to 3} (2x + 1)$.
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$7$. Direct substitution works for polynomial functions.
$7$. Direct substitution works for polynomial functions.
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Find $\text{lim}_{x \to 2} (x^2 - 4)/(x - 2)$. Is it continuous at $x = 2$?
Find $\text{lim}_{x \to 2} (x^2 - 4)/(x - 2)$. Is it continuous at $x = 2$?
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Limit is $4$, not continuous without $f(2)$ defined. Use L'Hôpital's rule or factor: limit is $4$, but function undefined.
Limit is $4$, not continuous without $f(2)$ defined. Use L'Hôpital's rule or factor: limit is $4$, but function undefined.
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Is $f(x) = \frac{1}{x}$ continuous at $x = 0$? Why or why not?
Is $f(x) = \frac{1}{x}$ continuous at $x = 0$? Why or why not?
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No, $f(x)$ is undefined at $x = 0$. Division by zero makes the function undefined at that point.
No, $f(x)$ is undefined at $x = 0$. Division by zero makes the function undefined at that point.
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Find the value of $c$ that makes $f(x) = cx + 1$ continuous at $x = 2$ if $f(2) = 5$.
Find the value of $c$ that makes $f(x) = cx + 1$ continuous at $x = 2$ if $f(2) = 5$.
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$c = 2$. Set $2c + 1 = 5$ and solve: $c = 2$.
$c = 2$. Set $2c + 1 = 5$ and solve: $c = 2$.
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What is the definition of continuity at a point $x = a$?
What is the definition of continuity at a point $x = a$?
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A function $f(x)$ is continuous at $x = a$ if $\text{lim}_{x \to a} f(x) = f(a)$. The limit must exist and equal the function value at that point.
A function $f(x)$ is continuous at $x = a$ if $\text{lim}_{x \to a} f(x) = f(a)$. The limit must exist and equal the function value at that point.
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Is $h(x) = \frac{1}{x-1}$ continuous at $x = 1$?
Is $h(x) = \frac{1}{x-1}$ continuous at $x = 1$?
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No, $h(x)$ is undefined at $x = 1$. Vertical asymptote creates an infinite discontinuity.
No, $h(x)$ is undefined at $x = 1$. Vertical asymptote creates an infinite discontinuity.
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State the Intermediate Value Theorem.
State the Intermediate Value Theorem.
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If $f(x)$ is continuous on $[a, b]$, $f(a) < N < f(b)$ then $\text{exists } c \text{ such that } f(c) = N$. Guarantees all intermediate values exist for continuous functions on closed intervals.
If $f(x)$ is continuous on $[a, b]$, $f(a) < N < f(b)$ then $\text{exists } c \text{ such that } f(c) = N$. Guarantees all intermediate values exist for continuous functions on closed intervals.
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What is the first step in confirming continuity at a point?
What is the first step in confirming continuity at a point?
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Verify $f(a)$ is defined. Cannot check continuity if the function value doesn't exist.
Verify $f(a)$ is defined. Cannot check continuity if the function value doesn't exist.
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What is the definition of a jump discontinuity?
What is the definition of a jump discontinuity?
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When $\lim_{x \to a^-} f(x) \neq \lim_{x \to a^+} f(x)$. Left and right approaches yield different limit values.
When $\lim_{x \to a^-} f(x) \neq \lim_{x \to a^+} f(x)$. Left and right approaches yield different limit values.
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Determine if $f(x) = \frac{x^2 - 9}{x - 3}$ is continuous at $x = 3$.
Determine if $f(x) = \frac{x^2 - 9}{x - 3}$ is continuous at $x = 3$.
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No, $f(x)$ has a removable discontinuity. Factor: $\frac{(x+3)(x-3)}{x-3} = x+3$, but $f(3)$ undefined.
No, $f(x)$ has a removable discontinuity. Factor: $\frac{(x+3)(x-3)}{x-3} = x+3$, but $f(3)$ undefined.
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What must be true for $f(x)$ to be continuous from the right at $x = a$?
What must be true for $f(x)$ to be continuous from the right at $x = a$?
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$\text{lim}_{x \to a^+} f(x) = f(a)$. Right-hand limit must equal the function value.
$\text{lim}_{x \to a^+} f(x) = f(a)$. Right-hand limit must equal the function value.
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What is a point of discontinuity?
What is a point of discontinuity?
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A point where $f(x)$ is not continuous. Where any of the three continuity conditions fail.
A point where $f(x)$ is not continuous. Where any of the three continuity conditions fail.
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If $f(x) = \frac{x^2 - 4}{x - 2}$, what value must $f(2)$ be for continuity at $x = 2$?
If $f(x) = \frac{x^2 - 4}{x - 2}$, what value must $f(2)$ be for continuity at $x = 2$?
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$4$. Factor and simplify: $\frac{(x+2)(x-2)}{x-2} = x+2$, so limit is $4$.
$4$. Factor and simplify: $\frac{(x+2)(x-2)}{x-2} = x+2$, so limit is $4$.
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What must be true about $\text{lim}{x \to a^-} f(x)$ and $\text{lim}{x \to a^+} f(x)$ for continuity at $x = a$?
What must be true about $\text{lim}{x \to a^-} f(x)$ and $\text{lim}{x \to a^+} f(x)$ for continuity at $x = a$?
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They must both exist and be equal to $f(a)$. Left and right limits must converge to the same value as the function.
They must both exist and be equal to $f(a)$. Left and right limits must converge to the same value as the function.
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Is $g(x) = \text{ln}(x)$ continuous for $x > 0$?
Is $g(x) = \text{ln}(x)$ continuous for $x > 0$?
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Yes, $g(x)$ is continuous for $x > 0$. Natural logarithm is continuous on its entire domain.
Yes, $g(x)$ is continuous for $x > 0$. Natural logarithm is continuous on its entire domain.
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What is the limit $\text{lim}_{x \to 2} 3x + 1$?
What is the limit $\text{lim}_{x \to 2} 3x + 1$?
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$7$. Direct substitution: $3(2) + 1 = 7$.
$7$. Direct substitution: $3(2) + 1 = 7$.
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Identify the type of discontinuity if $\text{lim}_{x \to a} f(x)$ does not exist.
Identify the type of discontinuity if $\text{lim}_{x \to a} f(x)$ does not exist.
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This is an infinite or jump discontinuity. When limits don't exist, the function has a break or approaches infinity.
This is an infinite or jump discontinuity. When limits don't exist, the function has a break or approaches infinity.
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If $f(x)$ is continuous on $[0, 1]$ and $f(0) = 3$, can $f(1) = 0$?
If $f(x)$ is continuous on $[0, 1]$ and $f(0) = 3$, can $f(1) = 0$?
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Yes, if $f(x)$ decreases continuously. IVT guarantees all values between $f(0)$ and $f(1)$ are achieved.
Yes, if $f(x)$ decreases continuously. IVT guarantees all values between $f(0)$ and $f(1)$ are achieved.
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If $\text{lim}_{x \to a} f(x) = L$ and $f(a) = L$, is $f(x)$ continuous at $x = a$?
If $\text{lim}_{x \to a} f(x) = L$ and $f(a) = L$, is $f(x)$ continuous at $x = a$?
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Yes, $f(x)$ is continuous at $x = a$. All three conditions for continuity are satisfied.
Yes, $f(x)$ is continuous at $x = a$. All three conditions for continuity are satisfied.
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Find $\text{lim}_{x \to 0} \frac{\text{sin}(x)}{x}$.
Find $\text{lim}_{x \to 0} \frac{\text{sin}(x)}{x}$.
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$1$. Standard trigonometric limit used in derivative of sine.
$1$. Standard trigonometric limit used in derivative of sine.
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For $f(x) = \frac{x^3 - 8}{x - 2}$, does $f(x)$ have a removable discontinuity at $x = 2$?
For $f(x) = \frac{x^3 - 8}{x - 2}$, does $f(x)$ have a removable discontinuity at $x = 2$?
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Yes, $f(x)$ has a removable discontinuity. Factor: $\frac{(x-2)(x^2+2x+4)}{x-2} = x^2+2x+4$, limit exists.
Yes, $f(x)$ has a removable discontinuity. Factor: $\frac{(x-2)(x^2+2x+4)}{x-2} = x^2+2x+4$, limit exists.
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What type of discontinuity is present if $f(x)$ is undefined at $x = a$?
What type of discontinuity is present if $f(x)$ is undefined at $x = a$?
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It could be a removable or infinite discontinuity. Depends on whether the function approaches infinity or has a finite limit.
It could be a removable or infinite discontinuity. Depends on whether the function approaches infinity or has a finite limit.
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State the condition for $f(x)$ to be continuous at an endpoint $a$ of $[a, b]$.
State the condition for $f(x)$ to be continuous at an endpoint $a$ of $[a, b]$.
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$\text{lim}_{x \to a^+} f(x) = f(a)$. Only right-hand continuity needed at left endpoint.
$\text{lim}_{x \to a^+} f(x) = f(a)$. Only right-hand continuity needed at left endpoint.
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Which theorem guarantees a root exists in $[a, b]$ if $f(a) \times f(b) < 0$?
Which theorem guarantees a root exists in $[a, b]$ if $f(a) \times f(b) < 0$?
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The Intermediate Value Theorem. Sign change guarantees a zero by IVT for continuous functions.
The Intermediate Value Theorem. Sign change guarantees a zero by IVT for continuous functions.
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What is the condition for continuity from the left at $x = a$?
What is the condition for continuity from the left at $x = a$?
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$\text{lim}_{x \to a^-} f(x) = f(a)$. Left-hand limit must equal the function value.
$\text{lim}_{x \to a^-} f(x) = f(a)$. Left-hand limit must equal the function value.
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For $g(x) = \frac{x^2 - 1}{x - 1}$, does $g(x)$ have a removable discontinuity at $x = 1$?
For $g(x) = \frac{x^2 - 1}{x - 1}$, does $g(x)$ have a removable discontinuity at $x = 1$?
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Yes, $g(x)$ has a removable discontinuity. Factor: $\frac{(x+1)(x-1)}{x-1} = x+1$, limit exists at $x = 1$.
Yes, $g(x)$ has a removable discontinuity. Factor: $\frac{(x+1)(x-1)}{x-1} = x+1$, limit exists at $x = 1$.
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Identify the discontinuity: $\text{lim}_{x \to a} f(x)$ exists but $f(a)$ is not defined.
Identify the discontinuity: $\text{lim}_{x \to a} f(x)$ exists but $f(a)$ is not defined.
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Removable discontinuity. The 'hole' can be filled by defining the function at that point.
Removable discontinuity. The 'hole' can be filled by defining the function at that point.
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If $f(x) = x^3$, is $f(x)$ continuous on $(-\text{\textinfty}, \text{\textinfty})$?
If $f(x) = x^3$, is $f(x)$ continuous on $(-\text{\textinfty}, \text{\textinfty})$?
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Yes, $f(x)$ is continuous everywhere. Polynomial functions are continuous everywhere in their domain.
Yes, $f(x)$ is continuous everywhere. Polynomial functions are continuous everywhere in their domain.
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What is a removable discontinuity?
What is a removable discontinuity?
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A point where a function is not defined, but the limit exists. Can be 'fixed' by defining or redefining the function at that point.
A point where a function is not defined, but the limit exists. Can be 'fixed' by defining or redefining the function at that point.
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What ensures a function is continuous on a closed interval $[a, b]$?
What ensures a function is continuous on a closed interval $[a, b]$?
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Continuous on $(a, b)$, right at $a$, left at $b$. Requires continuity at all interior points plus endpoint conditions.
Continuous on $(a, b)$, right at $a$, left at $b$. Requires continuity at all interior points plus endpoint conditions.
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