Is undefined and does not exist the same 2024?
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Emma Parker
Studied at Columbia University, Lives in New York City. Currently working as a marketing manager for a fashion brand.
As a mathematical expert, I'd like to clarify the concepts of "undefined" and "does not exist" in the context of limits in mathematics, particularly in calculus.
In mathematics, the statement "the limit is undefined" and "the limit does not exist" are often used interchangeably, but they have nuanced differences that are important to understand.
Firstly, let’s define what a limit is. A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. The limit of a function \( f(x) \) as \( x \) approaches \( c \) is said to be \( L \) if for every number \( \epsilon > 0 \), there exists a number \( \delta > 0 \) such that whenever \( 0 < |x - c| < \delta \), it follows that \( |f(x) - L| < \epsilon \). This is a formal definition that encapsulates the idea of a function approaching a certain value as its input gets arbitrarily close to a point.
Now, when we say that "the limit is undefined," it typically means that the function does not approach a specific value as \( x \) approaches \( c \). This could be due to several reasons:
1. Indeterminate Forms: The function may approach different values from different directions, leading to an indeterminate form such as \( 0/0 \) or \( \infty/\infty \), which does not settle on a single value.
2. Oscillation: The function may oscillate between different values without settling on a single value as \( x \) approaches \( c \).
3. Unbounded Behavior: As described in your reference, the function may get large without bound, meaning it goes to infinity or negative infinity, rather than approaching a finite value.
On the other hand, when we say that "the limit does not exist," it is a broader statement that encompasses all the scenarios where the limit is not a finite number. It includes the cases where the limit is undefined due to the reasons mentioned above, but it also includes the possibility that the function may not even approach infinity or negative infinity in a consistent manner.
To illustrate, consider the function \( f(x) = \sin(\frac{1}{x}) \) as \( x \) approaches \( 0 \). This function oscillates between -1 and 1 and does not approach any specific value. In this case, we say that the limit does not exist because there is no single value that \( f(x) \) is approaching. The limit is undefined because it does not approach a specific number, but rather oscillates indefinitely.
In summary, while "undefined" and "does not exist" are often used to mean the same thing in casual conversation, in a technical sense, "undefined" refers to a specific reason why the limit is not a finite number, whereas "does not exist" is a more general statement that includes all scenarios where the limit is not a finite number.
Now, let's proceed to the next step.
In mathematics, the statement "the limit is undefined" and "the limit does not exist" are often used interchangeably, but they have nuanced differences that are important to understand.
Firstly, let’s define what a limit is. A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. The limit of a function \( f(x) \) as \( x \) approaches \( c \) is said to be \( L \) if for every number \( \epsilon > 0 \), there exists a number \( \delta > 0 \) such that whenever \( 0 < |x - c| < \delta \), it follows that \( |f(x) - L| < \epsilon \). This is a formal definition that encapsulates the idea of a function approaching a certain value as its input gets arbitrarily close to a point.
Now, when we say that "the limit is undefined," it typically means that the function does not approach a specific value as \( x \) approaches \( c \). This could be due to several reasons:
1. Indeterminate Forms: The function may approach different values from different directions, leading to an indeterminate form such as \( 0/0 \) or \( \infty/\infty \), which does not settle on a single value.
2. Oscillation: The function may oscillate between different values without settling on a single value as \( x \) approaches \( c \).
3. Unbounded Behavior: As described in your reference, the function may get large without bound, meaning it goes to infinity or negative infinity, rather than approaching a finite value.
On the other hand, when we say that "the limit does not exist," it is a broader statement that encompasses all the scenarios where the limit is not a finite number. It includes the cases where the limit is undefined due to the reasons mentioned above, but it also includes the possibility that the function may not even approach infinity or negative infinity in a consistent manner.
To illustrate, consider the function \( f(x) = \sin(\frac{1}{x}) \) as \( x \) approaches \( 0 \). This function oscillates between -1 and 1 and does not approach any specific value. In this case, we say that the limit does not exist because there is no single value that \( f(x) \) is approaching. The limit is undefined because it does not approach a specific number, but rather oscillates indefinitely.
In summary, while "undefined" and "does not exist" are often used to mean the same thing in casual conversation, in a technical sense, "undefined" refers to a specific reason why the limit is not a finite number, whereas "does not exist" is a more general statement that includes all scenarios where the limit is not a finite number.
Now, let's proceed to the next step.
2024-06-17 00:26:26
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Works at the International Committee of the Red Cross, Lives in Geneva, Switzerland.
The answer to your question is that the limit is undefined if the limit does not exist as described by this technical definition. ... In this example the limit of f(x), as x approaches zero, does not exist since, as x approaches zero, the values of the function get large without bound.
2023-06-13 08:44:55
Benjamin Jackson
QuesHub.com delivers expert answers and knowledge to you.
The answer to your question is that the limit is undefined if the limit does not exist as described by this technical definition. ... In this example the limit of f(x), as x approaches zero, does not exist since, as x approaches zero, the values of the function get large without bound.
