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Questions and Answers
What is the definition of removable discontinuity?
What is the definition of removable discontinuity?
Which condition characterizes an essential discontinuity?
Which condition characterizes an essential discontinuity?
In the context of discontinuities, what is meant by one-sided limits?
In the context of discontinuities, what is meant by one-sided limits?
What condition must be met for a discontinuity to be removable?
What condition must be met for a discontinuity to be removable?
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When discussing discontinuities, what does it mean for a function to diverge?
When discussing discontinuities, what does it mean for a function to diverge?
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What is the domain of the function 𝑟𝑟2(𝑥𝑥)?
What is the domain of the function 𝑟𝑟2(𝑥𝑥)?
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If 𝑟𝑟2(𝑥𝑥) is in the domain of 𝑟𝑟1(𝑥𝑥), what is the relationship between 𝑟𝑟2(𝑥𝑥) and 𝑘𝑘?
If 𝑟𝑟2(𝑥𝑥) is in the domain of 𝑟𝑟1(𝑥𝑥), what is the relationship between 𝑟𝑟2(𝑥𝑥) and 𝑘𝑘?
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What type of functions are linear functions?
What type of functions are linear functions?
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What is the degree of a polynomial function denoted as deg(𝑓)?
What is the degree of a polynomial function denoted as deg(𝑓)?
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What kind of functions are defined by the greatest integer function?
What kind of functions are defined by the greatest integer function?
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If $\lim_{x \to b} g(x) = c$, then there exists $\delta_1 > 0$ such that if $0 < |x - b| < \delta_1$, then:
If $\lim_{x \to b} g(x) = c$, then there exists $\delta_1 > 0$ such that if $0 < |x - b| < \delta_1$, then:
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What notation represents the greatest integer less than or equal to 𝑥?
What notation represents the greatest integer less than or equal to 𝑥?
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If the limit of $g(x)$ as $x$ approaches 0 is a constant $c$, what can be concluded about the limit of $[f(x) + g(x)]$ as $x$ approaches $b$?
If the limit of $g(x)$ as $x$ approaches 0 is a constant $c$, what can be concluded about the limit of $[f(x) + g(x)]$ as $x$ approaches $b$?
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If $\lim_{x \to b} f(x) = 0$, then there exists a $\delta_2 > 0$ such that if $0 < |x - b| < \delta_2$, then:
If $\lim_{x \to b} f(x) = 0$, then there exists a $\delta_2 > 0$ such that if $0 < |x - b| < \delta_2$, then:
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In the case where $r$ is an odd number, what is the limit as $x$ approaches 0 of $f(x)$ when $f(x) = x^r$?
In the case where $r$ is an odd number, what is the limit as $x$ approaches 0 of $f(x)$ when $f(x) = x^r$?
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For which type of functions does Theorem 1.11 provide conclusions about the limits of the sum of two functions?
For which type of functions does Theorem 1.11 provide conclusions about the limits of the sum of two functions?
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When defining $\delta = \min(\delta_1, \delta_2)$, if $0 < |x - b| < \delta$, then:
When defining $\delta = \min(\delta_1, \delta_2)$, if $0 < |x - b| < \delta$, then:
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What is the limit of $f(x) \cdot g(x)$ as $x$ approaches $b$?
What is the limit of $f(x) \cdot g(x)$ as $x$ approaches $b$?
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What happens to the limit of $[f(x) + g(x)]$ as $x$ approaches $b$ if the limit of $f(x)$ is $-
fty$ according to Theorem 1.11?
What happens to the limit of $[f(x) + g(x)]$ as $x$ approaches $b$ if the limit of $f(x)$ is $- fty$ according to Theorem 1.11?
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What is the limit of $-f(x)$ as $x$ approaches $b$?
What is the limit of $-f(x)$ as $x$ approaches $b$?
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If a function $f(x)$ equals $x^r$ and $r$ is even, what can be said about the limit of $f(x)$ as $x$ approaches 0?
If a function $f(x)$ equals $x^r$ and $r$ is even, what can be said about the limit of $f(x)$ as $x$ approaches 0?
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What is the limit as $x$ approaches 0 of $g(x)$ when the limit of $f(x)$ is a constant according to Theorem 1.11?
What is the limit as $x$ approaches 0 of $g(x)$ when the limit of $f(x)$ is a constant according to Theorem 1.11?
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If $f(x) \to 0$ through positive values, what can be deduced about the function's behavior?
If $f(x) \to 0$ through positive values, what can be deduced about the function's behavior?
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If the limit of 𝑓𝑓(𝑥𝑥) as 𝑥𝑥 approaches 𝑏𝑏 is written as lim 𝑓𝑓(𝑥𝑥) = −∞, what does this mean?
If the limit of 𝑓𝑓(𝑥𝑥) as 𝑥𝑥 approaches 𝑏𝑏 is written as lim 𝑓𝑓(𝑥𝑥) = −∞, what does this mean?
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In the theorem presented, if the limit of 𝑔𝑔(𝑥𝑥) as 𝑥𝑥 approaches 𝑏𝑏 is a constant 𝑐𝑐 not equal to 0, what is the relationship between 𝑐𝑐 and the limits of 𝑓𝑓(𝑥𝑥) and 𝑔𝑔(𝑥𝑥)?
In the theorem presented, if the limit of 𝑔𝑔(𝑥𝑥) as 𝑥𝑥 approaches 𝑏𝑏 is a constant 𝑐𝑐 not equal to 0, what is the relationship between 𝑐𝑐 and the limits of 𝑓𝑓(𝑥𝑥) and 𝑔𝑔(𝑥𝑥)?
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Which scenario would lead to the limit lim = +∞ as 𝒙 approaches 𝒃?
Which scenario would lead to the limit lim = +∞ as 𝒙 approaches 𝒃?
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In the context of limits, what does it mean when a function 'decreases without bound'?
In the context of limits, what does it mean when a function 'decreases without bound'?
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If lim 𝒇(𝒙) = 0 and lim 𝒈(𝒙) = 𝒄, where 𝒄 is a non-zero constant, what can be concluded according to Theorem 1.9?
If lim 𝒇(𝒙) = 0 and lim 𝒈(𝒙) = 𝒄, where 𝒄 is a non-zero constant, what can be concluded according to Theorem 1.9?
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What does it imply if a function has a limit of +∞ as its input approaches a certain value?
What does it imply if a function has a limit of +∞ as its input approaches a certain value?
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