Is there a way to find this limit algebraically? $limlimits_{xto infty} frac{x}{sqrt{x^2 + 1}}$
I'm a Calculus I student and my teacher has given me a set of problems to solve with L'Hoptial's rule. Most of them have been pretty easy, but this one has me stumped.
$$limlimits_{xto infty} frac{x}{sqrt{x^2 + 1}}$$
You'll notice that using L'Hopital's rule flips the value of the top to the bottom. For example, using it once returns:
$$limlimits_{xto infty} frac{sqrt{x^2 + 1}}{x}$$
And doing it again returns you to the beginning:
$$limlimits_{xto infty} frac{x}{sqrt{x^2 + 1}}$$
I of course plugged it into my calculator to find the limit to evaluate to 1, but I was wondering if there was a better way to do this algebraically?
calculus
New contributor
add a comment |
I'm a Calculus I student and my teacher has given me a set of problems to solve with L'Hoptial's rule. Most of them have been pretty easy, but this one has me stumped.
$$limlimits_{xto infty} frac{x}{sqrt{x^2 + 1}}$$
You'll notice that using L'Hopital's rule flips the value of the top to the bottom. For example, using it once returns:
$$limlimits_{xto infty} frac{sqrt{x^2 + 1}}{x}$$
And doing it again returns you to the beginning:
$$limlimits_{xto infty} frac{x}{sqrt{x^2 + 1}}$$
I of course plugged it into my calculator to find the limit to evaluate to 1, but I was wondering if there was a better way to do this algebraically?
calculus
New contributor
6
$frac{x}{sqrt{x^2 + 1}}=frac{sqrt{x^2}}{sqrt{x^2+1}}=sqrt{frac{x^2}{x^2+1}}$
– Mason
yesterday
3
The "right" way to solve this is as Thomas Shelby wrote in his answer. However, you can almost intuitively see that the limit is $1$ even without using a calculator. Indeed, if $x$ is a huge positive number, say $10^{5863}$, the difference between $x^2$ and $x^2 + 1$ is extremely tiny. So the expression is essentially indistinguishable from ${xoversqrt{x^2}} = {xover x} = 1$.
– Andreas Rejbrand
yesterday
add a comment |
I'm a Calculus I student and my teacher has given me a set of problems to solve with L'Hoptial's rule. Most of them have been pretty easy, but this one has me stumped.
$$limlimits_{xto infty} frac{x}{sqrt{x^2 + 1}}$$
You'll notice that using L'Hopital's rule flips the value of the top to the bottom. For example, using it once returns:
$$limlimits_{xto infty} frac{sqrt{x^2 + 1}}{x}$$
And doing it again returns you to the beginning:
$$limlimits_{xto infty} frac{x}{sqrt{x^2 + 1}}$$
I of course plugged it into my calculator to find the limit to evaluate to 1, but I was wondering if there was a better way to do this algebraically?
calculus
New contributor
I'm a Calculus I student and my teacher has given me a set of problems to solve with L'Hoptial's rule. Most of them have been pretty easy, but this one has me stumped.
$$limlimits_{xto infty} frac{x}{sqrt{x^2 + 1}}$$
You'll notice that using L'Hopital's rule flips the value of the top to the bottom. For example, using it once returns:
$$limlimits_{xto infty} frac{sqrt{x^2 + 1}}{x}$$
And doing it again returns you to the beginning:
$$limlimits_{xto infty} frac{x}{sqrt{x^2 + 1}}$$
I of course plugged it into my calculator to find the limit to evaluate to 1, but I was wondering if there was a better way to do this algebraically?
calculus
calculus
New contributor
New contributor
edited yesterday
Blue
47.7k870151
47.7k870151
New contributor
asked yesterday
Jae Swanepoel
311
311
New contributor
New contributor
6
$frac{x}{sqrt{x^2 + 1}}=frac{sqrt{x^2}}{sqrt{x^2+1}}=sqrt{frac{x^2}{x^2+1}}$
– Mason
yesterday
3
The "right" way to solve this is as Thomas Shelby wrote in his answer. However, you can almost intuitively see that the limit is $1$ even without using a calculator. Indeed, if $x$ is a huge positive number, say $10^{5863}$, the difference between $x^2$ and $x^2 + 1$ is extremely tiny. So the expression is essentially indistinguishable from ${xoversqrt{x^2}} = {xover x} = 1$.
– Andreas Rejbrand
yesterday
add a comment |
6
$frac{x}{sqrt{x^2 + 1}}=frac{sqrt{x^2}}{sqrt{x^2+1}}=sqrt{frac{x^2}{x^2+1}}$
– Mason
yesterday
3
The "right" way to solve this is as Thomas Shelby wrote in his answer. However, you can almost intuitively see that the limit is $1$ even without using a calculator. Indeed, if $x$ is a huge positive number, say $10^{5863}$, the difference between $x^2$ and $x^2 + 1$ is extremely tiny. So the expression is essentially indistinguishable from ${xoversqrt{x^2}} = {xover x} = 1$.
– Andreas Rejbrand
yesterday
6
6
$frac{x}{sqrt{x^2 + 1}}=frac{sqrt{x^2}}{sqrt{x^2+1}}=sqrt{frac{x^2}{x^2+1}}$
– Mason
yesterday
$frac{x}{sqrt{x^2 + 1}}=frac{sqrt{x^2}}{sqrt{x^2+1}}=sqrt{frac{x^2}{x^2+1}}$
– Mason
yesterday
3
3
The "right" way to solve this is as Thomas Shelby wrote in his answer. However, you can almost intuitively see that the limit is $1$ even without using a calculator. Indeed, if $x$ is a huge positive number, say $10^{5863}$, the difference between $x^2$ and $x^2 + 1$ is extremely tiny. So the expression is essentially indistinguishable from ${xoversqrt{x^2}} = {xover x} = 1$.
– Andreas Rejbrand
yesterday
The "right" way to solve this is as Thomas Shelby wrote in his answer. However, you can almost intuitively see that the limit is $1$ even without using a calculator. Indeed, if $x$ is a huge positive number, say $10^{5863}$, the difference between $x^2$ and $x^2 + 1$ is extremely tiny. So the expression is essentially indistinguishable from ${xoversqrt{x^2}} = {xover x} = 1$.
– Andreas Rejbrand
yesterday
add a comment |
5 Answers
5
active
oldest
votes
Hint: Divide the numerator and denominator by $x $ and apply the limit.
$$frac{x}{sqrt{x^2 + 1}}=frac{1}{sqrt{1 + frac{1}{x^2}}}$$
5
In other words, actually do the division you see written down in $lim_{xtoinfty}frac{sqrt{x^2+1}}x$.
– Arthur
yesterday
add a comment |
Hint
Simply use $${xover x+1}={xover sqrt{x^2+2x+1}}<{xover sqrt{x^2+1}}<1$$for large enough $x>0$.
1
Mostafa.Very nice+.
– Peter Szilas
yesterday
But how so we know the limit is 1. Based on what you wrote couldnt the answer be 1/2
– Milan Stojanovic
yesterday
@PeterSzilas thank you!
– Mostafa Ayaz
yesterday
3
@MilanStojanovic No since $$lim_{xto infty}{xover x+1}=1$$and $$lim_{xto infty}1=1$$so the Squeeze theorem is applicable here....
– Mostafa Ayaz
yesterday
add a comment |
By your own reasoning, you have the following:
$$limlimits_{xto infty} frac{x}{sqrt{x^2 + 1}}=limlimits_{xto infty} frac{sqrt{x^2 + 1}}{x}$$
Now, the left side is clearly the reciprocal of the right side, so we have:
$$limlimits_{xto infty} frac{x}{sqrt{x^2 + 1}}=frac{1}{limlimits_{xto infty} frac{x}{sqrt{x^2 + 1}}}$$
(Note that doing this manipulation assumes that $limlimits_{xto infty} frac{x}{sqrt{x^2 + 1}}$ converges to a real number. However, you can use the first derivative to show this is an always increasing function and then use basic algebra to show that $frac{x}{sqrt{x^2 + 1}} < 1$ for all $xinBbb{R}$. Thus, because this is a bounded, always increasing function, the limit as $xto infty$ must converge to some real number, so our assumption in this manipulation is valid.)
Cross-multiply:
$$left(limlimits_{xto infty} frac{x}{sqrt{x^2 + 1}}right)^2=1$$
Take the square root:
$$limlimits_{xto infty} frac{x}{sqrt{x^2 + 1}}=pm 1$$
However, it is easy to show that $frac{x}{sqrt{x^2 + 1}}>0$ for all $x > 0$. Therefore, there's no way the limit can be a negative number like $-1$. Thus, the only possibility we have left is $+1$, so:
$$limlimits_{xto infty} frac{x}{sqrt{x^2 + 1}}=1$$
add a comment |
When computing the limit of rational functions, as is the case for $$lim_{x rightarrow infty} frac{x}{sqrt{x^2 +1}},$$ you want to divide the top and bottom by the highest degree in the denominator, which in this case is $x$. Since $x rightarrow +infty$, so $x$ is always positive (at least, near where we are worried about) I claim that $x = sqrt{x^2}$. So, if we divide the top and bottom by $x$, we get $$lim_{x rightarrow infty} frac{x}{sqrt{x^2 +1}} = lim_{x rightarrow infty} frac{1}{sqrt{1 + 1/x^2}}.$$ You should be able to compute the limit from here.
Whenever you see a monomial in the numerator with the square root of a polynomial in the denominator, you should consider this method. Of course, keep in mind that you'll have to tweak it slightly if $x rightarrow -infty$! Try to see if you can figure out what would change in that case.
New contributor
add a comment |
Set $x = sinh t$. We have
$$frac{x}{sqrt{x^2+1}}= frac{sinh t}{sqrt{1+sinh^2t}} = frac{sinh t}{cosh t} = tanh t$$
$x to infty$ is equivalent to $ttoinfty$ so $$lim_{xtoinfty} frac{x}{sqrt{x^2+1}} = lim_{ttoinfty} tanh t = lim_{ttoinfty}frac{e^t - e^{-t}}{e^t+e^{-t}} = lim_{ttoinfty}frac{e^{2t}-1}{e^{2t}+1} = 1$$
add a comment |
Your Answer
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5 Answers
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active
oldest
votes
5 Answers
5
active
oldest
votes
active
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votes
active
oldest
votes
Hint: Divide the numerator and denominator by $x $ and apply the limit.
$$frac{x}{sqrt{x^2 + 1}}=frac{1}{sqrt{1 + frac{1}{x^2}}}$$
5
In other words, actually do the division you see written down in $lim_{xtoinfty}frac{sqrt{x^2+1}}x$.
– Arthur
yesterday
add a comment |
Hint: Divide the numerator and denominator by $x $ and apply the limit.
$$frac{x}{sqrt{x^2 + 1}}=frac{1}{sqrt{1 + frac{1}{x^2}}}$$
5
In other words, actually do the division you see written down in $lim_{xtoinfty}frac{sqrt{x^2+1}}x$.
– Arthur
yesterday
add a comment |
Hint: Divide the numerator and denominator by $x $ and apply the limit.
$$frac{x}{sqrt{x^2 + 1}}=frac{1}{sqrt{1 + frac{1}{x^2}}}$$
Hint: Divide the numerator and denominator by $x $ and apply the limit.
$$frac{x}{sqrt{x^2 + 1}}=frac{1}{sqrt{1 + frac{1}{x^2}}}$$
edited yesterday
answered yesterday
Thomas Shelby
1,887217
1,887217
5
In other words, actually do the division you see written down in $lim_{xtoinfty}frac{sqrt{x^2+1}}x$.
– Arthur
yesterday
add a comment |
5
In other words, actually do the division you see written down in $lim_{xtoinfty}frac{sqrt{x^2+1}}x$.
– Arthur
yesterday
5
5
In other words, actually do the division you see written down in $lim_{xtoinfty}frac{sqrt{x^2+1}}x$.
– Arthur
yesterday
In other words, actually do the division you see written down in $lim_{xtoinfty}frac{sqrt{x^2+1}}x$.
– Arthur
yesterday
add a comment |
Hint
Simply use $${xover x+1}={xover sqrt{x^2+2x+1}}<{xover sqrt{x^2+1}}<1$$for large enough $x>0$.
1
Mostafa.Very nice+.
– Peter Szilas
yesterday
But how so we know the limit is 1. Based on what you wrote couldnt the answer be 1/2
– Milan Stojanovic
yesterday
@PeterSzilas thank you!
– Mostafa Ayaz
yesterday
3
@MilanStojanovic No since $$lim_{xto infty}{xover x+1}=1$$and $$lim_{xto infty}1=1$$so the Squeeze theorem is applicable here....
– Mostafa Ayaz
yesterday
add a comment |
Hint
Simply use $${xover x+1}={xover sqrt{x^2+2x+1}}<{xover sqrt{x^2+1}}<1$$for large enough $x>0$.
1
Mostafa.Very nice+.
– Peter Szilas
yesterday
But how so we know the limit is 1. Based on what you wrote couldnt the answer be 1/2
– Milan Stojanovic
yesterday
@PeterSzilas thank you!
– Mostafa Ayaz
yesterday
3
@MilanStojanovic No since $$lim_{xto infty}{xover x+1}=1$$and $$lim_{xto infty}1=1$$so the Squeeze theorem is applicable here....
– Mostafa Ayaz
yesterday
add a comment |
Hint
Simply use $${xover x+1}={xover sqrt{x^2+2x+1}}<{xover sqrt{x^2+1}}<1$$for large enough $x>0$.
Hint
Simply use $${xover x+1}={xover sqrt{x^2+2x+1}}<{xover sqrt{x^2+1}}<1$$for large enough $x>0$.
answered yesterday
Mostafa Ayaz
14.1k3937
14.1k3937
1
Mostafa.Very nice+.
– Peter Szilas
yesterday
But how so we know the limit is 1. Based on what you wrote couldnt the answer be 1/2
– Milan Stojanovic
yesterday
@PeterSzilas thank you!
– Mostafa Ayaz
yesterday
3
@MilanStojanovic No since $$lim_{xto infty}{xover x+1}=1$$and $$lim_{xto infty}1=1$$so the Squeeze theorem is applicable here....
– Mostafa Ayaz
yesterday
add a comment |
1
Mostafa.Very nice+.
– Peter Szilas
yesterday
But how so we know the limit is 1. Based on what you wrote couldnt the answer be 1/2
– Milan Stojanovic
yesterday
@PeterSzilas thank you!
– Mostafa Ayaz
yesterday
3
@MilanStojanovic No since $$lim_{xto infty}{xover x+1}=1$$and $$lim_{xto infty}1=1$$so the Squeeze theorem is applicable here....
– Mostafa Ayaz
yesterday
1
1
Mostafa.Very nice+.
– Peter Szilas
yesterday
Mostafa.Very nice+.
– Peter Szilas
yesterday
But how so we know the limit is 1. Based on what you wrote couldnt the answer be 1/2
– Milan Stojanovic
yesterday
But how so we know the limit is 1. Based on what you wrote couldnt the answer be 1/2
– Milan Stojanovic
yesterday
@PeterSzilas thank you!
– Mostafa Ayaz
yesterday
@PeterSzilas thank you!
– Mostafa Ayaz
yesterday
3
3
@MilanStojanovic No since $$lim_{xto infty}{xover x+1}=1$$and $$lim_{xto infty}1=1$$so the Squeeze theorem is applicable here....
– Mostafa Ayaz
yesterday
@MilanStojanovic No since $$lim_{xto infty}{xover x+1}=1$$and $$lim_{xto infty}1=1$$so the Squeeze theorem is applicable here....
– Mostafa Ayaz
yesterday
add a comment |
By your own reasoning, you have the following:
$$limlimits_{xto infty} frac{x}{sqrt{x^2 + 1}}=limlimits_{xto infty} frac{sqrt{x^2 + 1}}{x}$$
Now, the left side is clearly the reciprocal of the right side, so we have:
$$limlimits_{xto infty} frac{x}{sqrt{x^2 + 1}}=frac{1}{limlimits_{xto infty} frac{x}{sqrt{x^2 + 1}}}$$
(Note that doing this manipulation assumes that $limlimits_{xto infty} frac{x}{sqrt{x^2 + 1}}$ converges to a real number. However, you can use the first derivative to show this is an always increasing function and then use basic algebra to show that $frac{x}{sqrt{x^2 + 1}} < 1$ for all $xinBbb{R}$. Thus, because this is a bounded, always increasing function, the limit as $xto infty$ must converge to some real number, so our assumption in this manipulation is valid.)
Cross-multiply:
$$left(limlimits_{xto infty} frac{x}{sqrt{x^2 + 1}}right)^2=1$$
Take the square root:
$$limlimits_{xto infty} frac{x}{sqrt{x^2 + 1}}=pm 1$$
However, it is easy to show that $frac{x}{sqrt{x^2 + 1}}>0$ for all $x > 0$. Therefore, there's no way the limit can be a negative number like $-1$. Thus, the only possibility we have left is $+1$, so:
$$limlimits_{xto infty} frac{x}{sqrt{x^2 + 1}}=1$$
add a comment |
By your own reasoning, you have the following:
$$limlimits_{xto infty} frac{x}{sqrt{x^2 + 1}}=limlimits_{xto infty} frac{sqrt{x^2 + 1}}{x}$$
Now, the left side is clearly the reciprocal of the right side, so we have:
$$limlimits_{xto infty} frac{x}{sqrt{x^2 + 1}}=frac{1}{limlimits_{xto infty} frac{x}{sqrt{x^2 + 1}}}$$
(Note that doing this manipulation assumes that $limlimits_{xto infty} frac{x}{sqrt{x^2 + 1}}$ converges to a real number. However, you can use the first derivative to show this is an always increasing function and then use basic algebra to show that $frac{x}{sqrt{x^2 + 1}} < 1$ for all $xinBbb{R}$. Thus, because this is a bounded, always increasing function, the limit as $xto infty$ must converge to some real number, so our assumption in this manipulation is valid.)
Cross-multiply:
$$left(limlimits_{xto infty} frac{x}{sqrt{x^2 + 1}}right)^2=1$$
Take the square root:
$$limlimits_{xto infty} frac{x}{sqrt{x^2 + 1}}=pm 1$$
However, it is easy to show that $frac{x}{sqrt{x^2 + 1}}>0$ for all $x > 0$. Therefore, there's no way the limit can be a negative number like $-1$. Thus, the only possibility we have left is $+1$, so:
$$limlimits_{xto infty} frac{x}{sqrt{x^2 + 1}}=1$$
add a comment |
By your own reasoning, you have the following:
$$limlimits_{xto infty} frac{x}{sqrt{x^2 + 1}}=limlimits_{xto infty} frac{sqrt{x^2 + 1}}{x}$$
Now, the left side is clearly the reciprocal of the right side, so we have:
$$limlimits_{xto infty} frac{x}{sqrt{x^2 + 1}}=frac{1}{limlimits_{xto infty} frac{x}{sqrt{x^2 + 1}}}$$
(Note that doing this manipulation assumes that $limlimits_{xto infty} frac{x}{sqrt{x^2 + 1}}$ converges to a real number. However, you can use the first derivative to show this is an always increasing function and then use basic algebra to show that $frac{x}{sqrt{x^2 + 1}} < 1$ for all $xinBbb{R}$. Thus, because this is a bounded, always increasing function, the limit as $xto infty$ must converge to some real number, so our assumption in this manipulation is valid.)
Cross-multiply:
$$left(limlimits_{xto infty} frac{x}{sqrt{x^2 + 1}}right)^2=1$$
Take the square root:
$$limlimits_{xto infty} frac{x}{sqrt{x^2 + 1}}=pm 1$$
However, it is easy to show that $frac{x}{sqrt{x^2 + 1}}>0$ for all $x > 0$. Therefore, there's no way the limit can be a negative number like $-1$. Thus, the only possibility we have left is $+1$, so:
$$limlimits_{xto infty} frac{x}{sqrt{x^2 + 1}}=1$$
By your own reasoning, you have the following:
$$limlimits_{xto infty} frac{x}{sqrt{x^2 + 1}}=limlimits_{xto infty} frac{sqrt{x^2 + 1}}{x}$$
Now, the left side is clearly the reciprocal of the right side, so we have:
$$limlimits_{xto infty} frac{x}{sqrt{x^2 + 1}}=frac{1}{limlimits_{xto infty} frac{x}{sqrt{x^2 + 1}}}$$
(Note that doing this manipulation assumes that $limlimits_{xto infty} frac{x}{sqrt{x^2 + 1}}$ converges to a real number. However, you can use the first derivative to show this is an always increasing function and then use basic algebra to show that $frac{x}{sqrt{x^2 + 1}} < 1$ for all $xinBbb{R}$. Thus, because this is a bounded, always increasing function, the limit as $xto infty$ must converge to some real number, so our assumption in this manipulation is valid.)
Cross-multiply:
$$left(limlimits_{xto infty} frac{x}{sqrt{x^2 + 1}}right)^2=1$$
Take the square root:
$$limlimits_{xto infty} frac{x}{sqrt{x^2 + 1}}=pm 1$$
However, it is easy to show that $frac{x}{sqrt{x^2 + 1}}>0$ for all $x > 0$. Therefore, there's no way the limit can be a negative number like $-1$. Thus, the only possibility we have left is $+1$, so:
$$limlimits_{xto infty} frac{x}{sqrt{x^2 + 1}}=1$$
answered yesterday
Noble Mushtak
15.1k1735
15.1k1735
add a comment |
add a comment |
When computing the limit of rational functions, as is the case for $$lim_{x rightarrow infty} frac{x}{sqrt{x^2 +1}},$$ you want to divide the top and bottom by the highest degree in the denominator, which in this case is $x$. Since $x rightarrow +infty$, so $x$ is always positive (at least, near where we are worried about) I claim that $x = sqrt{x^2}$. So, if we divide the top and bottom by $x$, we get $$lim_{x rightarrow infty} frac{x}{sqrt{x^2 +1}} = lim_{x rightarrow infty} frac{1}{sqrt{1 + 1/x^2}}.$$ You should be able to compute the limit from here.
Whenever you see a monomial in the numerator with the square root of a polynomial in the denominator, you should consider this method. Of course, keep in mind that you'll have to tweak it slightly if $x rightarrow -infty$! Try to see if you can figure out what would change in that case.
New contributor
add a comment |
When computing the limit of rational functions, as is the case for $$lim_{x rightarrow infty} frac{x}{sqrt{x^2 +1}},$$ you want to divide the top and bottom by the highest degree in the denominator, which in this case is $x$. Since $x rightarrow +infty$, so $x$ is always positive (at least, near where we are worried about) I claim that $x = sqrt{x^2}$. So, if we divide the top and bottom by $x$, we get $$lim_{x rightarrow infty} frac{x}{sqrt{x^2 +1}} = lim_{x rightarrow infty} frac{1}{sqrt{1 + 1/x^2}}.$$ You should be able to compute the limit from here.
Whenever you see a monomial in the numerator with the square root of a polynomial in the denominator, you should consider this method. Of course, keep in mind that you'll have to tweak it slightly if $x rightarrow -infty$! Try to see if you can figure out what would change in that case.
New contributor
add a comment |
When computing the limit of rational functions, as is the case for $$lim_{x rightarrow infty} frac{x}{sqrt{x^2 +1}},$$ you want to divide the top and bottom by the highest degree in the denominator, which in this case is $x$. Since $x rightarrow +infty$, so $x$ is always positive (at least, near where we are worried about) I claim that $x = sqrt{x^2}$. So, if we divide the top and bottom by $x$, we get $$lim_{x rightarrow infty} frac{x}{sqrt{x^2 +1}} = lim_{x rightarrow infty} frac{1}{sqrt{1 + 1/x^2}}.$$ You should be able to compute the limit from here.
Whenever you see a monomial in the numerator with the square root of a polynomial in the denominator, you should consider this method. Of course, keep in mind that you'll have to tweak it slightly if $x rightarrow -infty$! Try to see if you can figure out what would change in that case.
New contributor
When computing the limit of rational functions, as is the case for $$lim_{x rightarrow infty} frac{x}{sqrt{x^2 +1}},$$ you want to divide the top and bottom by the highest degree in the denominator, which in this case is $x$. Since $x rightarrow +infty$, so $x$ is always positive (at least, near where we are worried about) I claim that $x = sqrt{x^2}$. So, if we divide the top and bottom by $x$, we get $$lim_{x rightarrow infty} frac{x}{sqrt{x^2 +1}} = lim_{x rightarrow infty} frac{1}{sqrt{1 + 1/x^2}}.$$ You should be able to compute the limit from here.
Whenever you see a monomial in the numerator with the square root of a polynomial in the denominator, you should consider this method. Of course, keep in mind that you'll have to tweak it slightly if $x rightarrow -infty$! Try to see if you can figure out what would change in that case.
New contributor
edited yesterday
Noble Mushtak
15.1k1735
15.1k1735
New contributor
answered yesterday
kkc
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Set $x = sinh t$. We have
$$frac{x}{sqrt{x^2+1}}= frac{sinh t}{sqrt{1+sinh^2t}} = frac{sinh t}{cosh t} = tanh t$$
$x to infty$ is equivalent to $ttoinfty$ so $$lim_{xtoinfty} frac{x}{sqrt{x^2+1}} = lim_{ttoinfty} tanh t = lim_{ttoinfty}frac{e^t - e^{-t}}{e^t+e^{-t}} = lim_{ttoinfty}frac{e^{2t}-1}{e^{2t}+1} = 1$$
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Set $x = sinh t$. We have
$$frac{x}{sqrt{x^2+1}}= frac{sinh t}{sqrt{1+sinh^2t}} = frac{sinh t}{cosh t} = tanh t$$
$x to infty$ is equivalent to $ttoinfty$ so $$lim_{xtoinfty} frac{x}{sqrt{x^2+1}} = lim_{ttoinfty} tanh t = lim_{ttoinfty}frac{e^t - e^{-t}}{e^t+e^{-t}} = lim_{ttoinfty}frac{e^{2t}-1}{e^{2t}+1} = 1$$
add a comment |
Set $x = sinh t$. We have
$$frac{x}{sqrt{x^2+1}}= frac{sinh t}{sqrt{1+sinh^2t}} = frac{sinh t}{cosh t} = tanh t$$
$x to infty$ is equivalent to $ttoinfty$ so $$lim_{xtoinfty} frac{x}{sqrt{x^2+1}} = lim_{ttoinfty} tanh t = lim_{ttoinfty}frac{e^t - e^{-t}}{e^t+e^{-t}} = lim_{ttoinfty}frac{e^{2t}-1}{e^{2t}+1} = 1$$
Set $x = sinh t$. We have
$$frac{x}{sqrt{x^2+1}}= frac{sinh t}{sqrt{1+sinh^2t}} = frac{sinh t}{cosh t} = tanh t$$
$x to infty$ is equivalent to $ttoinfty$ so $$lim_{xtoinfty} frac{x}{sqrt{x^2+1}} = lim_{ttoinfty} tanh t = lim_{ttoinfty}frac{e^t - e^{-t}}{e^t+e^{-t}} = lim_{ttoinfty}frac{e^{2t}-1}{e^{2t}+1} = 1$$
answered yesterday
mechanodroid
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Jae Swanepoel is a new contributor. Be nice, and check out our Code of Conduct.
Jae Swanepoel is a new contributor. Be nice, and check out our Code of Conduct.
Jae Swanepoel is a new contributor. Be nice, and check out our Code of Conduct.
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6
$frac{x}{sqrt{x^2 + 1}}=frac{sqrt{x^2}}{sqrt{x^2+1}}=sqrt{frac{x^2}{x^2+1}}$
– Mason
yesterday
3
The "right" way to solve this is as Thomas Shelby wrote in his answer. However, you can almost intuitively see that the limit is $1$ even without using a calculator. Indeed, if $x$ is a huge positive number, say $10^{5863}$, the difference between $x^2$ and $x^2 + 1$ is extremely tiny. So the expression is essentially indistinguishable from ${xoversqrt{x^2}} = {xover x} = 1$.
– Andreas Rejbrand
yesterday