Mixing
Should one use “tends to” or “equals” when dealing with infinity?
$begingroup$
Is this perfectly valid:
$$lim_{xto0}frac{1}{x}=infty tag{1}$$
or should I use:
$$frac{1}{x}toinftyquadtext{as}quad xto0 tag{2}$$
or likewise:
$$lim_{xto0}frac{1}{x}toinfty tag{3}$$
are all valid? I always got taught you should never put infinity in an equation or as an answer, rather you should say "f(x) tends to infinity" because nothing can "equal" infinity, however my textbook is using the first example, likewise with many resources I've looked at online.
Cheers!
calculus limits notation
edited Jan 18 at 15:33
Blue
48.6k870156
asked Jan 18 at 15:31
Adam BromileyAdam Bromiley
281110
$endgroup$
add a comment |
$begingroup$
Is this perfectly valid:
$$lim_{xto0}frac{1}{x}=infty tag{1}$$
or should I use:
$$frac{1}{x}toinftyquadtext{as}quad xto0 tag{2}$$
or likewise:
$$lim_{xto0}frac{1}{x}toinfty tag{3}$$
are all valid? I always got taught you should never put infinity in an equation or as an answer, rather you should say "f(x) tends to infinity" because nothing can "equal" infinity, however my textbook is using the first example, likewise with many resources I've looked at online.
Cheers!
calculus limits notation
edited Jan 18 at 15:33
Blue
48.6k870156
asked Jan 18 at 15:31
Adam BromileyAdam Bromiley
281110
$endgroup$
add a comment |
$begingroup$
Is this perfectly valid:
$$lim_{xto0}frac{1}{x}=infty tag{1}$$
or should I use:
$$frac{1}{x}toinftyquadtext{as}quad xto0 tag{2}$$
or likewise:
$$lim_{xto0}frac{1}{x}toinfty tag{3}$$
are all valid? I always got taught you should never put infinity in an equation or as an answer, rather you should say "f(x) tends to infinity" because nothing can "equal" infinity, however my textbook is using the first example, likewise with many resources I've looked at online.
Cheers!
calculus limits notation
edited Jan 18 at 15:33
Blue
48.6k870156
asked Jan 18 at 15:31
Adam BromileyAdam Bromiley
281110
$endgroup$
Is this perfectly valid:
$$lim_{xto0}frac{1}{x}=infty tag{1}$$
or should I use:
$$frac{1}{x}toinftyquadtext{as}quad xto0 tag{2}$$
or likewise:
$$lim_{xto0}frac{1}{x}toinfty tag{3}$$
are all valid? I always got taught you should never put infinity in an equation or as an answer, rather you should say "f(x) tends to infinity" because nothing can "equal" infinity, however my textbook is using the first example, likewise with many resources I've looked at online.
Cheers!
calculus limits notation
calculus limits notation
edited Jan 18 at 15:33
Blue
48.6k870156
asked Jan 18 at 15:31
Adam BromileyAdam Bromiley
281110
edited Jan 18 at 15:33
Blue
48.6k870156
asked Jan 18 at 15:31
Adam BromileyAdam Bromiley
281110
edited Jan 18 at 15:33
Blue
48.6k870156
edited Jan 18 at 15:33
Blue
48.6k870156
edited Jan 18 at 15:33
Blue
48.6k870156
48.6k870156
asked Jan 18 at 15:31
Adam BromileyAdam Bromiley
281110
asked Jan 18 at 15:31
Adam BromileyAdam Bromiley
281110
asked Jan 18 at 15:31
Adam BromileyAdam Bromiley
281110
281110
add a comment |
add a comment |
4 Answers
4
active
oldest
votes
$begingroup$
The first notation and the second are both common, the third is never used. Yes, we define $lim_{xto x_0}f(x)=infty$ and we can use the $infty$ symbol in equations, appropriately. However, the equations themselves are in fact incorrect. The limits $lim_{xto 0}frac{1}{x}$ does not exist, since if we approach $0$ from the left and the right we get different results: $lim_{xto 0^+}frac{1}{x}=infty$ and $lim_{xto 0^-}frac{1}{x}=-infty$, where the former means approach $0$ from the right and the latter from the left. The limits are different and hence saying $lim_{xto 0}frac{1}{x}=infty$ is incorrect.
answered Jan 18 at 15:39
Yuval GatYuval Gat
572213
$endgroup$
$begingroup$
Perfect, thank you!
$endgroup$
– Adam Bromiley
Jan 19 at 16:05
add a comment |
$begingroup$
Good question!
Essentially we have that
$$frac 1x to infty text { as } xto 0^+$$
Where $to$ has an English meaning of "tends to"
And we also have that:
$$lim_{xto 0^+}{frac 1x}=infty$$
"The limit as x tends to 0 from the positive side of 1 over x is infinity"
In short, use "tends to" when not using the limit notation, use equals when you are.
answered Jan 18 at 15:37
Rhys HughesRhys Hughes
6,9341530
$endgroup$
add a comment |
$begingroup$
You were taught wrong. "$lim_{x to 0} f(x) = infty$" has a precise meaning, which is the same as "$f(x) to infty$ as $x to 0$". On the other hand, you should never say $lim_{x to 0} f(x) to infty$.
Your particular example, though, is wrong. $lim_{x to 0} 1/x$ doesn't exist,
as it involves both positive and negative values of $x$.
$lim_{x to 0+} 1/x = +infty$, but $lim_{x to 0-} 1/x = -infty$.
answered Jan 18 at 15:37
Robert IsraelRobert Israel
325k23214468
$endgroup$
add a comment |
$begingroup$
The first two are valid and mean the same thing - the third makes very little sense. Don't ever write the third, please.
Regarding whether nothing can equal infinity: WE have a definition of what $lim_{xto0}1/x=infty $ means - there are no infinities in that definition.
answered Jan 18 at 15:34
David C. UllrichDavid C. Ullrich
61k43994
$endgroup$
add a comment |
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4 Answers
4
active
oldest
votes
4 Answers
4
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The first notation and the second are both common, the third is never used. Yes, we define $lim_{xto x_0}f(x)=infty$ and we can use the $infty$ symbol in equations, appropriately. However, the equations themselves are in fact incorrect. The limits $lim_{xto 0}frac{1}{x}$ does not exist, since if we approach $0$ from the left and the right we get different results: $lim_{xto 0^+}frac{1}{x}=infty$ and $lim_{xto 0^-}frac{1}{x}=-infty$, where the former means approach $0$ from the right and the latter from the left. The limits are different and hence saying $lim_{xto 0}frac{1}{x}=infty$ is incorrect.
answered Jan 18 at 15:39
Yuval GatYuval Gat
572213
$endgroup$
$begingroup$
Perfect, thank you!
$endgroup$
– Adam Bromiley
Jan 19 at 16:05
add a comment |
$begingroup$
The first notation and the second are both common, the third is never used. Yes, we define $lim_{xto x_0}f(x)=infty$ and we can use the $infty$ symbol in equations, appropriately. However, the equations themselves are in fact incorrect. The limits $lim_{xto 0}frac{1}{x}$ does not exist, since if we approach $0$ from the left and the right we get different results: $lim_{xto 0^+}frac{1}{x}=infty$ and $lim_{xto 0^-}frac{1}{x}=-infty$, where the former means approach $0$ from the right and the latter from the left. The limits are different and hence saying $lim_{xto 0}frac{1}{x}=infty$ is incorrect.
answered Jan 18 at 15:39
Yuval GatYuval Gat
572213
$endgroup$
$begingroup$
Perfect, thank you!
$endgroup$
– Adam Bromiley
Jan 19 at 16:05
add a comment |
$begingroup$
The first notation and the second are both common, the third is never used. Yes, we define $lim_{xto x_0}f(x)=infty$ and we can use the $infty$ symbol in equations, appropriately. However, the equations themselves are in fact incorrect. The limits $lim_{xto 0}frac{1}{x}$ does not exist, since if we approach $0$ from the left and the right we get different results: $lim_{xto 0^+}frac{1}{x}=infty$ and $lim_{xto 0^-}frac{1}{x}=-infty$, where the former means approach $0$ from the right and the latter from the left. The limits are different and hence saying $lim_{xto 0}frac{1}{x}=infty$ is incorrect.
answered Jan 18 at 15:39
Yuval GatYuval Gat
572213
$endgroup$
The first notation and the second are both common, the third is never used. Yes, we define $lim_{xto x_0}f(x)=infty$ and we can use the $infty$ symbol in equations, appropriately. However, the equations themselves are in fact incorrect. The limits $lim_{xto 0}frac{1}{x}$ does not exist, since if we approach $0$ from the left and the right we get different results: $lim_{xto 0^+}frac{1}{x}=infty$ and $lim_{xto 0^-}frac{1}{x}=-infty$, where the former means approach $0$ from the right and the latter from the left. The limits are different and hence saying $lim_{xto 0}frac{1}{x}=infty$ is incorrect.
answered Jan 18 at 15:39
Yuval GatYuval Gat
572213
answered Jan 18 at 15:39
Yuval GatYuval Gat
572213
answered Jan 18 at 15:39
Yuval GatYuval Gat
572213
answered Jan 18 at 15:39
Yuval GatYuval Gat
572213
572213
$begingroup$
Perfect, thank you!
$endgroup$
– Adam Bromiley
Jan 19 at 16:05
add a comment |
$begingroup$
Perfect, thank you!
$endgroup$
– Adam Bromiley
Jan 19 at 16:05
$begingroup$
Perfect, thank you!
$endgroup$
– Adam Bromiley
Jan 19 at 16:05
$begingroup$
Perfect, thank you!
$endgroup$
– Adam Bromiley
Jan 19 at 16:05
add a comment |
$begingroup$
Good question!
Essentially we have that
$$frac 1x to infty text { as } xto 0^+$$
Where $to$ has an English meaning of "tends to"
And we also have that:
$$lim_{xto 0^+}{frac 1x}=infty$$
"The limit as x tends to 0 from the positive side of 1 over x is infinity"
In short, use "tends to" when not using the limit notation, use equals when you are.
answered Jan 18 at 15:37
Rhys HughesRhys Hughes
6,9341530
$endgroup$
add a comment |
$begingroup$
Good question!
Essentially we have that
$$frac 1x to infty text { as } xto 0^+$$
Where $to$ has an English meaning of "tends to"
And we also have that:
$$lim_{xto 0^+}{frac 1x}=infty$$
"The limit as x tends to 0 from the positive side of 1 over x is infinity"
In short, use "tends to" when not using the limit notation, use equals when you are.
answered Jan 18 at 15:37
Rhys HughesRhys Hughes
6,9341530
$endgroup$
add a comment |
$begingroup$
Good question!
Essentially we have that
$$frac 1x to infty text { as } xto 0^+$$
Where $to$ has an English meaning of "tends to"
And we also have that:
$$lim_{xto 0^+}{frac 1x}=infty$$
"The limit as x tends to 0 from the positive side of 1 over x is infinity"
In short, use "tends to" when not using the limit notation, use equals when you are.
answered Jan 18 at 15:37
Rhys HughesRhys Hughes
6,9341530
$endgroup$
Good question!
Essentially we have that
$$frac 1x to infty text { as } xto 0^+$$
Where $to$ has an English meaning of "tends to"
And we also have that:
$$lim_{xto 0^+}{frac 1x}=infty$$
"The limit as x tends to 0 from the positive side of 1 over x is infinity"
In short, use "tends to" when not using the limit notation, use equals when you are.
answered Jan 18 at 15:37
Rhys HughesRhys Hughes
6,9341530
answered Jan 18 at 15:37
Rhys HughesRhys Hughes
6,9341530
answered Jan 18 at 15:37
Rhys HughesRhys Hughes
6,9341530
answered Jan 18 at 15:37
Rhys HughesRhys Hughes
6,9341530
6,9341530
add a comment |
add a comment |
$begingroup$
You were taught wrong. "$lim_{x to 0} f(x) = infty$" has a precise meaning, which is the same as "$f(x) to infty$ as $x to 0$". On the other hand, you should never say $lim_{x to 0} f(x) to infty$.
Your particular example, though, is wrong. $lim_{x to 0} 1/x$ doesn't exist,
as it involves both positive and negative values of $x$.
$lim_{x to 0+} 1/x = +infty$, but $lim_{x to 0-} 1/x = -infty$.
answered Jan 18 at 15:37
Robert IsraelRobert Israel
325k23214468
$endgroup$
add a comment |
$begingroup$
You were taught wrong. "$lim_{x to 0} f(x) = infty$" has a precise meaning, which is the same as "$f(x) to infty$ as $x to 0$". On the other hand, you should never say $lim_{x to 0} f(x) to infty$.
Your particular example, though, is wrong. $lim_{x to 0} 1/x$ doesn't exist,
as it involves both positive and negative values of $x$.
$lim_{x to 0+} 1/x = +infty$, but $lim_{x to 0-} 1/x = -infty$.
answered Jan 18 at 15:37
Robert IsraelRobert Israel
325k23214468
$endgroup$
add a comment |
$begingroup$
You were taught wrong. "$lim_{x to 0} f(x) = infty$" has a precise meaning, which is the same as "$f(x) to infty$ as $x to 0$". On the other hand, you should never say $lim_{x to 0} f(x) to infty$.
Your particular example, though, is wrong. $lim_{x to 0} 1/x$ doesn't exist,
as it involves both positive and negative values of $x$.
$lim_{x to 0+} 1/x = +infty$, but $lim_{x to 0-} 1/x = -infty$.
answered Jan 18 at 15:37
Robert IsraelRobert Israel
325k23214468
$endgroup$
You were taught wrong. "$lim_{x to 0} f(x) = infty$" has a precise meaning, which is the same as "$f(x) to infty$ as $x to 0$". On the other hand, you should never say $lim_{x to 0} f(x) to infty$.
Your particular example, though, is wrong. $lim_{x to 0} 1/x$ doesn't exist,
as it involves both positive and negative values of $x$.
$lim_{x to 0+} 1/x = +infty$, but $lim_{x to 0-} 1/x = -infty$.
answered Jan 18 at 15:37
Robert IsraelRobert Israel
325k23214468
answered Jan 18 at 15:37
Robert IsraelRobert Israel
325k23214468
answered Jan 18 at 15:37
Robert IsraelRobert Israel
325k23214468
answered Jan 18 at 15:37
Robert IsraelRobert Israel
325k23214468
325k23214468
add a comment |
add a comment |
$begingroup$
The first two are valid and mean the same thing - the third makes very little sense. Don't ever write the third, please.
Regarding whether nothing can equal infinity: WE have a definition of what $lim_{xto0}1/x=infty $ means - there are no infinities in that definition.
answered Jan 18 at 15:34
David C. UllrichDavid C. Ullrich
61k43994
$endgroup$
add a comment |
$begingroup$
The first two are valid and mean the same thing - the third makes very little sense. Don't ever write the third, please.
Regarding whether nothing can equal infinity: WE have a definition of what $lim_{xto0}1/x=infty $ means - there are no infinities in that definition.
answered Jan 18 at 15:34
David C. UllrichDavid C. Ullrich
61k43994
$endgroup$
add a comment |
$begingroup$
The first two are valid and mean the same thing - the third makes very little sense. Don't ever write the third, please.
Regarding whether nothing can equal infinity: WE have a definition of what $lim_{xto0}1/x=infty $ means - there are no infinities in that definition.
answered Jan 18 at 15:34
David C. UllrichDavid C. Ullrich
61k43994
$endgroup$
The first two are valid and mean the same thing - the third makes very little sense. Don't ever write the third, please.
Regarding whether nothing can equal infinity: WE have a definition of what $lim_{xto0}1/x=infty $ means - there are no infinities in that definition.
answered Jan 18 at 15:34
David C. UllrichDavid C. Ullrich
61k43994
answered Jan 18 at 15:34
David C. UllrichDavid C. Ullrich
61k43994
answered Jan 18 at 15:34
David C. UllrichDavid C. Ullrich
61k43994
answered Jan 18 at 15:34
David C. UllrichDavid C. Ullrich
61k43994
61k43994
add a comment |
add a comment |
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