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How can we define the limit of a constant?
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Wikipedia says:
In mathematics, a limit is the value that a function(or sequence) "approaches" as the input (or index) "approaches" some value.
What if the function was a constant?! A constant is fixed and will not approach anything, so how would we define the limit of a constant?
calculus limits
asked 1 hour ago
مÙÂØÂَمَّدْ
163
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add a comment |Â
up vote
2
down vote
favorite
Wikipedia says:
In mathematics, a limit is the value that a function(or sequence) "approaches" as the input (or index) "approaches" some value.
What if the function was a constant?! A constant is fixed and will not approach anything, so how would we define the limit of a constant?
calculus limits
asked 1 hour ago
مÙÂØÂَمَّدْ
163
New contributor
مÙÂØÂَمَّدْ is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Limit of WHAT? There is no "input" to a constant. It is nonsense to write $lim_x to x_0 pi$, for instance.
– David G. Stork
1 hour ago
2
David I would interpret your limit to have value $pi$.
– M_B
1 hour ago
3
@DavidG.Stork It's not nonsense to write $lim_x to x_0 pi$, but it is trivial. Here, $pi = f(x)$ is a constant function, so the definition is perfectly valid here.
– JavaMan
57 mins ago
The value of a function $f(x) $ may equal its limit $L$ as $xto a$. Check for example $f(x) =xsin (1/x)$ as $xto 0$. For a constant function the function value always equals its limits.
– Paramanand Singh
29 mins ago
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Wikipedia says:
In mathematics, a limit is the value that a function(or sequence) "approaches" as the input (or index) "approaches" some value.
What if the function was a constant?! A constant is fixed and will not approach anything, so how would we define the limit of a constant?
calculus limits
asked 1 hour ago
مÙÂØÂَمَّدْ
163
New contributor
مÙÂØÂَمَّدْ is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Wikipedia says:
In mathematics, a limit is the value that a function(or sequence) "approaches" as the input (or index) "approaches" some value.
What if the function was a constant?! A constant is fixed and will not approach anything, so how would we define the limit of a constant?
calculus limits
calculus limits
asked 1 hour ago
مÙÂØÂَمَّدْ
163
New contributor
مÙÂØÂَمَّدْ is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 1 hour ago
مÙÂØÂَمَّدْ
163
New contributor
مÙÂØÂَمَّدْ is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 1 hour ago
مÙÂØÂَمَّدْ
163
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مÙÂØÂَمَّدْ is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 1 hour ago
مÙÂØÂَمَّدْ
163
asked 1 hour ago
مÙÂØÂَمَّدْ
163
163
New contributor
مÙÂØÂَمَّدْ is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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New contributor
مÙÂØÂَمَّدْ is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
مÙÂØÂَمَّدْ is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Limit of WHAT? There is no "input" to a constant. It is nonsense to write $lim_x to x_0 pi$, for instance.
– David G. Stork
1 hour ago
2
David I would interpret your limit to have value $pi$.
– M_B
1 hour ago
3
@DavidG.Stork It's not nonsense to write $lim_x to x_0 pi$, but it is trivial. Here, $pi = f(x)$ is a constant function, so the definition is perfectly valid here.
– JavaMan
57 mins ago
The value of a function $f(x) $ may equal its limit $L$ as $xto a$. Check for example $f(x) =xsin (1/x)$ as $xto 0$. For a constant function the function value always equals its limits.
– Paramanand Singh
29 mins ago
add a comment |Â
Limit of WHAT? There is no "input" to a constant. It is nonsense to write $lim_x to x_0 pi$, for instance.
– David G. Stork
1 hour ago
2
David I would interpret your limit to have value $pi$.
– M_B
1 hour ago
3
@DavidG.Stork It's not nonsense to write $lim_x to x_0 pi$, but it is trivial. Here, $pi = f(x)$ is a constant function, so the definition is perfectly valid here.
– JavaMan
57 mins ago
The value of a function $f(x) $ may equal its limit $L$ as $xto a$. Check for example $f(x) =xsin (1/x)$ as $xto 0$. For a constant function the function value always equals its limits.
– Paramanand Singh
29 mins ago
Limit of WHAT? There is no "input" to a constant. It is nonsense to write $lim_x to x_0 pi$, for instance.
– David G. Stork
1 hour ago
Limit of WHAT? There is no "input" to a constant. It is nonsense to write $lim_x to x_0 pi$, for instance.
– David G. Stork
1 hour ago
2
2
David I would interpret your limit to have value $pi$.
– M_B
1 hour ago
David I would interpret your limit to have value $pi$.
– M_B
1 hour ago
3
3
@DavidG.Stork It's not nonsense to write $lim_x to x_0 pi$, but it is trivial. Here, $pi = f(x)$ is a constant function, so the definition is perfectly valid here.
– JavaMan
57 mins ago
@DavidG.Stork It's not nonsense to write $lim_x to x_0 pi$, but it is trivial. Here, $pi = f(x)$ is a constant function, so the definition is perfectly valid here.
– JavaMan
57 mins ago
The value of a function $f(x) $ may equal its limit $L$ as $xto a$. Check for example $f(x) =xsin (1/x)$ as $xto 0$. For a constant function the function value always equals its limits.
– Paramanand Singh
29 mins ago
The value of a function $f(x) $ may equal its limit $L$ as $xto a$. Check for example $f(x) =xsin (1/x)$ as $xto 0$. For a constant function the function value always equals its limits.
– Paramanand Singh
29 mins ago
add a comment |Â
2 Answers
2
active
oldest
votes
up vote
3
down vote
Let $f(x)=c$, where $c$ is some constant number, and suppose that the domain of $f$ is the real numbers. Then we can take the limit as $x$ approaches some value, say for example $$lim_xrightarrow0f(x).$$
Then by definition $f(x)=c hspace0.1cm$ for all $x$, so in particular $$lim_xrightarrow0f(x)=c.$$
Notice that the value of the function $f$ is constant, but the value of the independent variable $x$ changes.
answered 1 hour ago
M_B
360110
Okay, according to the quote "the input (or index)" here which is $x$ can approach but the function $f$ is still a constant that won't approach anything; so, it is not right to say that the function $f$ approaches $c$ as $x$ approaches $0$
– Ù…ÙÂØÂَمَّدْ
26 mins ago
It is still right to say f approaches c, even when f is constantly c. Draw a picture of the graph of $f(x)=3$. It is a horizontal line at height 3. Taking the limit as x approaches 0 is like sliding your finger towards 0 from the left or right along the graph. The x values get closer and closer to 0 and the f values “get closer and closer to 3â€Â, even though in this case the f values are always 3.
– M_B
3 mins ago
add a comment |Â
up vote
2
down vote
Your quote isn't a definition of a limit, but an English language description of what it computes.
If you look at the actual definition, such as the usual "epsilon-delta" definition, you'll see that it handles constant functions just fine, and in fact you have
$$ lim_x to a c = c $$
So, what has happened here is that there is a miscommunication — the meaning intended by the author of this wikipedia passage is not the meaning inferred by you the reader.
While you could decide that the author used sloppy language, or you could chalk up the whole thing to the imprecision of the English language, I think the following will be more useful:
- You need to refine your intuition about what "approach" means
Natural language tends to be exclusive of overly simple or degenerate cases — e.g. if you were to say "I live within 50 miles of Paris" in everyday conversation, the listener would assume that you don't live in Paris because it's expected that you would have said so.
Technical language tend to be more inclusive unless stated otherwise — e.g. that someone living in Paris does indeed live within 50 miles of Paris.
answered 26 mins ago
Hurkyl
109k9114255
Thank you, but how can I define "approach"?
– Ù…ÙÂØÂَمَّدْ
22 mins ago
@مÙÂØÂَمَّدْ Look up $(epsilon-delta)$ definition. This is the usual definition used for defining a limit. You can find a review of this idea is here: khanacademy.org/math/calculus-all-old/… Temporarily forget the Wikipedia definition, and try to see if the idea of a limit makes sense under the "new" definition.
– Dair
17 mins ago
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
Let $f(x)=c$, where $c$ is some constant number, and suppose that the domain of $f$ is the real numbers. Then we can take the limit as $x$ approaches some value, say for example $$lim_xrightarrow0f(x).$$
Then by definition $f(x)=c hspace0.1cm$ for all $x$, so in particular $$lim_xrightarrow0f(x)=c.$$
Notice that the value of the function $f$ is constant, but the value of the independent variable $x$ changes.
answered 1 hour ago
M_B
360110
Okay, according to the quote "the input (or index)" here which is $x$ can approach but the function $f$ is still a constant that won't approach anything; so, it is not right to say that the function $f$ approaches $c$ as $x$ approaches $0$
– Ù…ÙÂØÂَمَّدْ
26 mins ago
It is still right to say f approaches c, even when f is constantly c. Draw a picture of the graph of $f(x)=3$. It is a horizontal line at height 3. Taking the limit as x approaches 0 is like sliding your finger towards 0 from the left or right along the graph. The x values get closer and closer to 0 and the f values “get closer and closer to 3â€Â, even though in this case the f values are always 3.
– M_B
3 mins ago
add a comment |Â
up vote
3
down vote
Let $f(x)=c$, where $c$ is some constant number, and suppose that the domain of $f$ is the real numbers. Then we can take the limit as $x$ approaches some value, say for example $$lim_xrightarrow0f(x).$$
Then by definition $f(x)=c hspace0.1cm$ for all $x$, so in particular $$lim_xrightarrow0f(x)=c.$$
Notice that the value of the function $f$ is constant, but the value of the independent variable $x$ changes.
answered 1 hour ago
M_B
360110
Okay, according to the quote "the input (or index)" here which is $x$ can approach but the function $f$ is still a constant that won't approach anything; so, it is not right to say that the function $f$ approaches $c$ as $x$ approaches $0$
– Ù…ÙÂØÂَمَّدْ
26 mins ago
It is still right to say f approaches c, even when f is constantly c. Draw a picture of the graph of $f(x)=3$. It is a horizontal line at height 3. Taking the limit as x approaches 0 is like sliding your finger towards 0 from the left or right along the graph. The x values get closer and closer to 0 and the f values “get closer and closer to 3â€Â, even though in this case the f values are always 3.
– M_B
3 mins ago
add a comment |Â
up vote
3
down vote
up vote
3
down vote
Let $f(x)=c$, where $c$ is some constant number, and suppose that the domain of $f$ is the real numbers. Then we can take the limit as $x$ approaches some value, say for example $$lim_xrightarrow0f(x).$$
Then by definition $f(x)=c hspace0.1cm$ for all $x$, so in particular $$lim_xrightarrow0f(x)=c.$$
Notice that the value of the function $f$ is constant, but the value of the independent variable $x$ changes.
answered 1 hour ago
M_B
360110
Let $f(x)=c$, where $c$ is some constant number, and suppose that the domain of $f$ is the real numbers. Then we can take the limit as $x$ approaches some value, say for example $$lim_xrightarrow0f(x).$$
Then by definition $f(x)=c hspace0.1cm$ for all $x$, so in particular $$lim_xrightarrow0f(x)=c.$$
Notice that the value of the function $f$ is constant, but the value of the independent variable $x$ changes.
answered 1 hour ago
M_B
360110
answered 1 hour ago
M_B
360110
answered 1 hour ago
M_B
360110
answered 1 hour ago
M_B
360110
360110
Okay, according to the quote "the input (or index)" here which is $x$ can approach but the function $f$ is still a constant that won't approach anything; so, it is not right to say that the function $f$ approaches $c$ as $x$ approaches $0$
– Ù…ÙÂØÂَمَّدْ
26 mins ago
It is still right to say f approaches c, even when f is constantly c. Draw a picture of the graph of $f(x)=3$. It is a horizontal line at height 3. Taking the limit as x approaches 0 is like sliding your finger towards 0 from the left or right along the graph. The x values get closer and closer to 0 and the f values “get closer and closer to 3â€Â, even though in this case the f values are always 3.
– M_B
3 mins ago
add a comment |Â
Okay, according to the quote "the input (or index)" here which is $x$ can approach but the function $f$ is still a constant that won't approach anything; so, it is not right to say that the function $f$ approaches $c$ as $x$ approaches $0$
– Ù…ÙÂØÂَمَّدْ
26 mins ago
It is still right to say f approaches c, even when f is constantly c. Draw a picture of the graph of $f(x)=3$. It is a horizontal line at height 3. Taking the limit as x approaches 0 is like sliding your finger towards 0 from the left or right along the graph. The x values get closer and closer to 0 and the f values “get closer and closer to 3â€Â, even though in this case the f values are always 3.
– M_B
3 mins ago
Okay, according to the quote "the input (or index)" here which is $x$ can approach but the function $f$ is still a constant that won't approach anything; so, it is not right to say that the function $f$ approaches $c$ as $x$ approaches $0$
– Ù…ÙÂØÂَمَّدْ
26 mins ago
Okay, according to the quote "the input (or index)" here which is $x$ can approach but the function $f$ is still a constant that won't approach anything; so, it is not right to say that the function $f$ approaches $c$ as $x$ approaches $0$
– Ù…ÙÂØÂَمَّدْ
26 mins ago
It is still right to say f approaches c, even when f is constantly c. Draw a picture of the graph of $f(x)=3$. It is a horizontal line at height 3. Taking the limit as x approaches 0 is like sliding your finger towards 0 from the left or right along the graph. The x values get closer and closer to 0 and the f values “get closer and closer to 3â€Â, even though in this case the f values are always 3.
– M_B
3 mins ago
It is still right to say f approaches c, even when f is constantly c. Draw a picture of the graph of $f(x)=3$. It is a horizontal line at height 3. Taking the limit as x approaches 0 is like sliding your finger towards 0 from the left or right along the graph. The x values get closer and closer to 0 and the f values “get closer and closer to 3â€Â, even though in this case the f values are always 3.
– M_B
3 mins ago
add a comment |Â
up vote
2
down vote
Your quote isn't a definition of a limit, but an English language description of what it computes.
If you look at the actual definition, such as the usual "epsilon-delta" definition, you'll see that it handles constant functions just fine, and in fact you have
$$ lim_x to a c = c $$
So, what has happened here is that there is a miscommunication — the meaning intended by the author of this wikipedia passage is not the meaning inferred by you the reader.
While you could decide that the author used sloppy language, or you could chalk up the whole thing to the imprecision of the English language, I think the following will be more useful:
- You need to refine your intuition about what "approach" means
Natural language tends to be exclusive of overly simple or degenerate cases — e.g. if you were to say "I live within 50 miles of Paris" in everyday conversation, the listener would assume that you don't live in Paris because it's expected that you would have said so.
Technical language tend to be more inclusive unless stated otherwise — e.g. that someone living in Paris does indeed live within 50 miles of Paris.
answered 26 mins ago
Hurkyl
109k9114255
Thank you, but how can I define "approach"?
– Ù…ÙÂØÂَمَّدْ
22 mins ago
@مÙÂØÂَمَّدْ Look up $(epsilon-delta)$ definition. This is the usual definition used for defining a limit. You can find a review of this idea is here: khanacademy.org/math/calculus-all-old/… Temporarily forget the Wikipedia definition, and try to see if the idea of a limit makes sense under the "new" definition.
– Dair
17 mins ago
add a comment |Â
up vote
2
down vote
Your quote isn't a definition of a limit, but an English language description of what it computes.
If you look at the actual definition, such as the usual "epsilon-delta" definition, you'll see that it handles constant functions just fine, and in fact you have
$$ lim_x to a c = c $$
So, what has happened here is that there is a miscommunication — the meaning intended by the author of this wikipedia passage is not the meaning inferred by you the reader.
While you could decide that the author used sloppy language, or you could chalk up the whole thing to the imprecision of the English language, I think the following will be more useful:
- You need to refine your intuition about what "approach" means
Natural language tends to be exclusive of overly simple or degenerate cases — e.g. if you were to say "I live within 50 miles of Paris" in everyday conversation, the listener would assume that you don't live in Paris because it's expected that you would have said so.
Technical language tend to be more inclusive unless stated otherwise — e.g. that someone living in Paris does indeed live within 50 miles of Paris.
answered 26 mins ago
Hurkyl
109k9114255
Thank you, but how can I define "approach"?
– Ù…ÙÂØÂَمَّدْ
22 mins ago
@مÙÂØÂَمَّدْ Look up $(epsilon-delta)$ definition. This is the usual definition used for defining a limit. You can find a review of this idea is here: khanacademy.org/math/calculus-all-old/… Temporarily forget the Wikipedia definition, and try to see if the idea of a limit makes sense under the "new" definition.
– Dair
17 mins ago
add a comment |Â
up vote
2
down vote
up vote
2
down vote
Your quote isn't a definition of a limit, but an English language description of what it computes.
If you look at the actual definition, such as the usual "epsilon-delta" definition, you'll see that it handles constant functions just fine, and in fact you have
$$ lim_x to a c = c $$
So, what has happened here is that there is a miscommunication — the meaning intended by the author of this wikipedia passage is not the meaning inferred by you the reader.
While you could decide that the author used sloppy language, or you could chalk up the whole thing to the imprecision of the English language, I think the following will be more useful:
- You need to refine your intuition about what "approach" means
Natural language tends to be exclusive of overly simple or degenerate cases — e.g. if you were to say "I live within 50 miles of Paris" in everyday conversation, the listener would assume that you don't live in Paris because it's expected that you would have said so.
Technical language tend to be more inclusive unless stated otherwise — e.g. that someone living in Paris does indeed live within 50 miles of Paris.
answered 26 mins ago
Hurkyl
109k9114255
Your quote isn't a definition of a limit, but an English language description of what it computes.
If you look at the actual definition, such as the usual "epsilon-delta" definition, you'll see that it handles constant functions just fine, and in fact you have
$$ lim_x to a c = c $$
So, what has happened here is that there is a miscommunication — the meaning intended by the author of this wikipedia passage is not the meaning inferred by you the reader.
While you could decide that the author used sloppy language, or you could chalk up the whole thing to the imprecision of the English language, I think the following will be more useful:
- You need to refine your intuition about what "approach" means
Natural language tends to be exclusive of overly simple or degenerate cases — e.g. if you were to say "I live within 50 miles of Paris" in everyday conversation, the listener would assume that you don't live in Paris because it's expected that you would have said so.
Technical language tend to be more inclusive unless stated otherwise — e.g. that someone living in Paris does indeed live within 50 miles of Paris.
answered 26 mins ago
Hurkyl
109k9114255
edited 21 mins ago
answered 26 mins ago
Hurkyl
109k9114255
answered 26 mins ago
Hurkyl
109k9114255
answered 26 mins ago
Hurkyl
109k9114255
109k9114255
Thank you, but how can I define "approach"?
– Ù…ÙÂØÂَمَّدْ
22 mins ago
@مÙÂØÂَمَّدْ Look up $(epsilon-delta)$ definition. This is the usual definition used for defining a limit. You can find a review of this idea is here: khanacademy.org/math/calculus-all-old/… Temporarily forget the Wikipedia definition, and try to see if the idea of a limit makes sense under the "new" definition.
– Dair
17 mins ago
add a comment |Â
Thank you, but how can I define "approach"?
– Ù…ÙÂØÂَمَّدْ
22 mins ago
@مÙÂØÂَمَّدْ Look up $(epsilon-delta)$ definition. This is the usual definition used for defining a limit. You can find a review of this idea is here: khanacademy.org/math/calculus-all-old/… Temporarily forget the Wikipedia definition, and try to see if the idea of a limit makes sense under the "new" definition.
– Dair
17 mins ago
Thank you, but how can I define "approach"?
– Ù…ÙÂØÂَمَّدْ
22 mins ago
Thank you, but how can I define "approach"?
– Ù…ÙÂØÂَمَّدْ
22 mins ago
@مÙÂØÂَمَّدْ Look up $(epsilon-delta)$ definition. This is the usual definition used for defining a limit. You can find a review of this idea is here: khanacademy.org/math/calculus-all-old/… Temporarily forget the Wikipedia definition, and try to see if the idea of a limit makes sense under the "new" definition.
– Dair
17 mins ago
@مÙÂØÂَمَّدْ Look up $(epsilon-delta)$ definition. This is the usual definition used for defining a limit. You can find a review of this idea is here: khanacademy.org/math/calculus-all-old/… Temporarily forget the Wikipedia definition, and try to see if the idea of a limit makes sense under the "new" definition.
– Dair
17 mins ago
add a comment |Â
مÙÂØÂَمَّدْ is a new contributor. Be nice, and check out our Code of Conduct.
مÙÂØÂَمَّدْ is a new contributor. Be nice, and check out our Code of Conduct.
مÙÂØÂَمَّدْ is a new contributor. Be nice, and check out our Code of Conduct.
مÙÂØÂَمَّدْ is a new contributor. Be nice, and check out our Code of Conduct.
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Limit of WHAT? There is no "input" to a constant. It is nonsense to write $lim_x to x_0 pi$, for instance.
– David G. Stork
1 hour ago
2
David I would interpret your limit to have value $pi$.
– M_B
1 hour ago
3
@DavidG.Stork It's not nonsense to write $lim_x to x_0 pi$, but it is trivial. Here, $pi = f(x)$ is a constant function, so the definition is perfectly valid here.
– JavaMan
57 mins ago
The value of a function $f(x) $ may equal its limit $L$ as $xto a$. Check for example $f(x) =xsin (1/x)$ as $xto 0$. For a constant function the function value always equals its limits.
– Paramanand Singh
29 mins ago