Expert: Paul Klarreich - 10/9/2007

Question
I had Calculus many, many years ago.  As my son takes Calculus, I'm working
some of the problems to see what I can remember.  I'm stuck on finding the
limit as x approaches 0 of    {1/[x times square root of (1 + x)]} - 1/x
I can graph it and see the result that way but can't seem to figure out a way to
make an meaningful interpretation otherwise.
    Thanks in advance!


Answer
Questioner:   John
Category:  Calculus
Private:  No
 
Subject:  Limits
Question:  I had Calculus many, many years ago.  As my son takes Calculus, I'm working some of the problems to see what I can remember.  I'm stuck on finding the
limit as x approaches 0 of    {1/[x times square root of (1 + x)]} - 1/x
I can graph it and see the result that way but can't seem to figure out a way to
make an meaningful interpretation otherwise.
   Thanks in advance!

....................................
Hi, John,

You are lucky -- you and your son are still at the ages where you know more math than he does.  Enjoy it -- it might not last long.

For your example:
=======================================================
WARNING: USE COURIER FONT TO VIEW THIS
===================================================

    1            1
------------- - -----
x sqrt(1 + x)     x

All you really need is a little algebra.  First combine the two fractions using your LCD:

1 - sqrt(1 + x)
---------------
x sqrt(1 + x)

Now rationalize the numerator.

1 - sqrt(1 + x)  1 + sqrt(1 + x)
---------------  ---------------
x sqrt(1 + x)   1 + sqrt(1 + x)

      1 - (1 + x)
-----------------------------
x sqrt(1 + x)(1 + sqrt(1 + x))

Simplify:

      1 - 1 - x
-----------------------------
x sqrt(1 + x)(1 + sqrt(1 + x))

          - x
-----------------------------
x sqrt(1 + x)(1 + sqrt(1 + x))

          - 1
-----------------------------
sqrt(1 + x)(1 + sqrt(1 + x))

Now that your pesky factor of x has cancelled, you can let  x -> 0

          - 1
-----------------------------
sqrt(1 + 0)(1 + sqrt(1 + 0))

       - 1
----------------------
sqrt(1)(1 + sqrt(1))

  - 1
----------
1(1 + 1)

- 1
----
 2