Is there a way to find this limit algebraically? $limlimits_{xto infty} frac{x}{sqrt{x^2 + 1}}$












6














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?










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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


















6














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?










share|cite|improve this question









New contributor




Jae Swanepoel is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
















  • 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








6







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?










share|cite|improve this question









New contributor




Jae Swanepoel is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











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






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Jae Swanepoel is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











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share|cite|improve this question








edited yesterday









Blue

47.7k870151




47.7k870151






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asked yesterday









Jae Swanepoel

311




311




New contributor




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New contributor





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Check out our Code of Conduct.






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Check out our Code of Conduct.








  • 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




    $frac{x}{sqrt{x^2 + 1}}=frac{sqrt{x^2}}{sqrt{x^2+1}}=sqrt{frac{x^2}{x^2+1}}$
    – 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












5 Answers
5






active

oldest

votes


















12














Hint: Divide the numerator and denominator by $x $ and apply the limit.



$$frac{x}{sqrt{x^2 + 1}}=frac{1}{sqrt{1 + frac{1}{x^2}}}$$






share|cite|improve this answer



















  • 5




    In other words, actually do the division you see written down in $lim_{xtoinfty}frac{sqrt{x^2+1}}x$.
    – Arthur
    yesterday





















11














Hint



Simply use $${xover x+1}={xover sqrt{x^2+2x+1}}<{xover sqrt{x^2+1}}<1$$for large enough $x>0$.






share|cite|improve this answer

















  • 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



















6














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:
$$limlimits_{xto infty} frac{x}{sqrt{x^2 + 1}}=1$$






share|cite|improve this answer





























    4














    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.






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      0














      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 $$lim_{xtoinfty} frac{x}{sqrt{x^2+1}} = lim_{ttoinfty} tanh t = lim_{ttoinfty}frac{e^t - e^{-t}}{e^t+e^{-t}} = lim_{ttoinfty}frac{e^{2t}-1}{e^{2t}+1} = 1$$






      share|cite|improve this answer





















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        5 Answers
        5






        active

        oldest

        votes








        5 Answers
        5






        active

        oldest

        votes









        active

        oldest

        votes






        active

        oldest

        votes









        12














        Hint: Divide the numerator and denominator by $x $ and apply the limit.



        $$frac{x}{sqrt{x^2 + 1}}=frac{1}{sqrt{1 + frac{1}{x^2}}}$$






        share|cite|improve this answer



















        • 5




          In other words, actually do the division you see written down in $lim_{xtoinfty}frac{sqrt{x^2+1}}x$.
          – Arthur
          yesterday


















        12














        Hint: Divide the numerator and denominator by $x $ and apply the limit.



        $$frac{x}{sqrt{x^2 + 1}}=frac{1}{sqrt{1 + frac{1}{x^2}}}$$






        share|cite|improve this answer



















        • 5




          In other words, actually do the division you see written down in $lim_{xtoinfty}frac{sqrt{x^2+1}}x$.
          – Arthur
          yesterday
















        12












        12








        12






        Hint: Divide the numerator and denominator by $x $ and apply the limit.



        $$frac{x}{sqrt{x^2 + 1}}=frac{1}{sqrt{1 + frac{1}{x^2}}}$$






        share|cite|improve this answer














        Hint: Divide the numerator and denominator by $x $ and apply the limit.



        $$frac{x}{sqrt{x^2 + 1}}=frac{1}{sqrt{1 + frac{1}{x^2}}}$$







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        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
















        • 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













        11














        Hint



        Simply use $${xover x+1}={xover sqrt{x^2+2x+1}}<{xover sqrt{x^2+1}}<1$$for large enough $x>0$.






        share|cite|improve this answer

















        • 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
















        11














        Hint



        Simply use $${xover x+1}={xover sqrt{x^2+2x+1}}<{xover sqrt{x^2+1}}<1$$for large enough $x>0$.






        share|cite|improve this answer

















        • 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














        11












        11








        11






        Hint



        Simply use $${xover x+1}={xover sqrt{x^2+2x+1}}<{xover sqrt{x^2+1}}<1$$for large enough $x>0$.






        share|cite|improve this answer












        Hint



        Simply use $${xover x+1}={xover sqrt{x^2+2x+1}}<{xover sqrt{x^2+1}}<1$$for large enough $x>0$.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        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














        • 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











        6














        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:
        $$limlimits_{xto infty} frac{x}{sqrt{x^2 + 1}}=1$$






        share|cite|improve this answer


























          6














          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:
          $$limlimits_{xto infty} frac{x}{sqrt{x^2 + 1}}=1$$






          share|cite|improve this answer
























            6












            6








            6






            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:
            $$limlimits_{xto infty} frac{x}{sqrt{x^2 + 1}}=1$$






            share|cite|improve this answer












            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:
            $$limlimits_{xto infty} frac{x}{sqrt{x^2 + 1}}=1$$







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            answered yesterday









            Noble Mushtak

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                4














                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.






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                  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.






                  share|cite|improve this answer










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                    4












                    4








                    4






                    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.






                    share|cite|improve this answer










                    New contributor




                    kkc is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                    Check out our Code of Conduct.









                    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.







                    share|cite|improve this answer










                    New contributor




                    kkc is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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                    share|cite|improve this answer








                    edited yesterday









                    Noble Mushtak

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                    answered yesterday









                    kkc

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                        0














                        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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                          0














                          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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                            0












                            0








                            0






                            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 $$lim_{xtoinfty} frac{x}{sqrt{x^2+1}} = lim_{ttoinfty} tanh t = lim_{ttoinfty}frac{e^t - e^{-t}}{e^t+e^{-t}} = lim_{ttoinfty}frac{e^{2t}-1}{e^{2t}+1} = 1$$






                            share|cite|improve this answer












                            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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                            answered yesterday









                            mechanodroid

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