29 Questions
What is the definition of removable discontinuity?
Exists at the point
Which condition characterizes an essential discontinuity?
Limit does not exist
In the context of discontinuities, what is meant by one-sided limits?
Limits approaching from one direction
What condition must be met for a discontinuity to be removable?
Function value at the point exists
When discussing discontinuities, what does it mean for a function to diverge?
Function value approaches infinity
What is the domain of the function 𝑟𝑟2(𝑥𝑥)?
[3, +∞)
If 𝑟𝑟2(𝑥𝑥) is in the domain of 𝑟𝑟1(𝑥𝑥), what is the relationship between 𝑟𝑟2(𝑥𝑥) and 𝑘𝑘?
𝜋𝜋2𝑥𝑥 − 3 ≠ 𝑘𝑘𝑘𝑘 +
What type of functions are linear functions?
Real-valued functions of the form 𝑓(𝑥) = 𝑚 + 𝑏
What is the degree of a polynomial function denoted as deg(𝑓)?
The highest power of 𝑥 in the function
What kind of functions are defined by the greatest integer function?
Step functions
If $\lim_{x \to b} g(x) = c$, then there exists $\delta_1 > 0$ such that if $0 < |x - b| < \delta_1$, then:
$|g(x) - c| < 2c$
What notation represents the greatest integer less than or equal to 𝑥?
[x]
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 $b$?
It will be $+ fty$ if the limit of $f(x)$ is $+ fty$.
If $\lim_{x \to b} f(x) = 0$, then there exists a $\delta_2 > 0$ such that if $0 < |x - b| < \delta_2$, then:
$1 < f(x) < 2N$
In the case where $r$ is an odd number, what is the limit as $x$ approaches 0 of $f(x)$ when $f(x) = x^r$?
$+ fty$
For which type of functions does Theorem 1.11 provide conclusions about the limits of the sum of two functions?
When one function has a limit at a constant value.
When defining $\delta = \min(\delta_1, \delta_2)$, if $0 < |x - b| < \delta$, then:
$g(x) > 2N$
What is the limit of $f(x) \cdot g(x)$ as $x$ approaches $b$?
$+\infty$
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?
It will be $- fty$.
What is the limit of $-f(x)$ as $x$ approaches $b$?
$0$
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?
$0$
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?
$0$
If $f(x) \to 0$ through positive values, what can be deduced about the function's behavior?
$f(x)$ oscillates between negative and positive values
If the limit of 𝑓𝑓(𝑥𝑥) as 𝑥𝑥 approaches 𝑏𝑏 is written as lim 𝑓𝑓(𝑥𝑥) = −∞, what does this mean?
As 𝑥𝑥 gets closer to 𝑏𝑏, 𝑓𝑓(𝑥𝑥) decreases without bound.
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)?
If 𝑐𝑐 > 0 and 𝑓𝑓(𝑥𝑥) approaches 0 through positive values, then the limit of 𝑔𝑔(𝑥𝑥) is positive infinity.
Which scenario would lead to the limit lim = +∞ as 𝒙 approaches 𝒃?
If the function 𝒇(𝒙) approaches 0 through positive values and the constant is positive.
In the context of limits, what does it mean when a function 'decreases without bound'?
The function approaches negative infinity or decreases indefinitely as the input approaches a specific value.
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 the constant is positive and 𝒈(𝒙) approaches 0 through positive values, then lim = +∞.
What does it imply if a function has a limit of +∞ as its input approaches a certain value?
The function will increase indefinitely as its input gets closer to that value.
This quiz covers the concept of limits and absolute values in Calculus, focusing on understanding the conditions for a function approaching a constant value. Topics include determining the existence of a delta and epsilon, and analyzing the behavior of functions near a specific point.
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