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MathGroup Archive 2001

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Re: Limit involving square root

  • To: mathgroup at smc.vnet.net
  • Subject: [mg30210] Re: Limit involving square root
  • From: "Carl K. Woll" <carlw at u.washington.edu>
  • Date: Thu, 2 Aug 2001 03:16:04 -0400 (EDT)
  • Organization: University of Washington
  • References: <9k880u$4vh$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Hi Hugh,

I don't know what's going wrong in your example, but there are two possible
remedies. First, there is a Calculus`Limit` package where Limit has been
enhanced. If you load this package and then try your limit, you will get the
correct result:

In[1]:=
<<Calculus`Limit`

In[2]:=
Limit[1+x-Sqrt[x^2-10x+9],x->Infinity]

Out[2]=
6

On the other hand, I prefer a different approach, using the Series command
as follows:

In[1]:=
Series[x-Sqrt[x^2-10x+9],{x,Infinity,3}]
Series[1+x-Sqrt[x^2-10x+9],{x,Infinity,3}]

Out[1]=
    8   40     1 3
5 + - + -- + O[-]
    x    2     x
        x

Out[2]=
    8   40     1 3
6 + - + -- + O[-]
    x    2     x
        x

As you can see (if you are using a fixed font), the Series command works
fine without introducing the Calculus`Limit` package.

Carl Woll
Physics Dept
U of Washington

"Hugh Goyder" <goyder at rmcs.cranfield.ac.uk> wrote in message
news:9k880u$4vh$1 at smc.vnet.net...
> Dear Mathgroup,
>
> Below I take the limit of a function and then the limit of 1 plus the same
> function. A plot of the function shows that the first result, (5), is
> correct but the second, (1), is wrong. (Should be 6.) What's happening?
>
> In[1]:=
> $Version
>
> Out[1]=
> 4.1 for Microsoft Windows (November 2, 2000)
>
> In[2]:=
> Limit[x - Sqrt[9 - 10 x + x^2],x -> Infinity]
>
> Out[2]=
> 5
>
> In[3]:=
> Limit[1 + x - Sqrt[9 - 10 x + x^2],x -> Infinity]
>
> Out[3]=
> 1
>
>
>
> I also note that using Series to expand about infinity does give the
> correct answers.
>
>
> Thanks
>
> Hugh Goyder
>




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