Simplifying an Equation

  • #1
Technically this is a calculus problem I'm working on, but I'm just having problems with the Algebra portion.

If I have:

[tex](\frac{1}{x\sqrt{1+x}} - \frac{1}{x})[/tex]

How can I simply this so that I can substitute in 0 for x?
 

Answers and Replies

  • #2
1,056
0
You want to get this in a form for the use of L'Hospital's Rule: [tex]\frac{1-\sqrt{1+x}}{x(\sqrt{1+x})}[/tex]

In this form we see that as [tex]x\rightarrow0[/tex] the quotient is undefined, so we can differentiate and simplify.
 
Last edited:
  • #3
We haven't gone into differentiation or anything like that, is there another way?

Actually, the problem that I'm trying to figure out is

lim
x -> 0 of the expression above.


edit: For clarification - it's not for homework, it's just a problem I'm trying to figure out.
 
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  • #4
1,056
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I don't know any other way to do this problem. This is how you do it using the Calculus. You differentiate and get:

[tex]\frac{[-2\sqrt{1+x}]^-1}{(2+3x)[2\sqrt{1+x}]^-1}=\frac{-1}{2+3x}\rightarrow \frac{-1}{2} ...as.... x \rightarrow 0[/tex]
 
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  • #5
Fredrik
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The substitution

[tex]t=\sqrt{1+x}[/tex]

simplifies the function to

[tex]-\frac{1}{(1+t)t}[/tex]

The limit of this as t goes to 1 is -1/2.
 
  • #6
1,056
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Fredrik said:
The substitution

[tex]t=\sqrt{1+x}[/tex]

simplifies the function to

[tex]-\frac{1}{(1+t)t}[/tex]

The limit of this as t goes to 1 is -1/2.

That looks like a better way!
 
  • #7
shmoe
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To add yet another way, rationalize the numerator. Multiply

[tex]\frac{1-\sqrt{1+x}}{x(\sqrt{1+x})}[/tex]

by

[tex]\frac{1+\sqrt{1+x}}{1+\sqrt{1+x}}[/tex]

to get

[tex]\frac{-x}{x(\sqrt{1+x})(1+\sqrt{1+x})}[/tex]

and go from there.
 
  • #8
Gokul43201
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Science Advisor
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And finally see that #5 and #7 are really doing the same thing.

They's both making use of the fact that x can be factored as [itex]-(1-\sqrt{1+x})(1+\sqrt{1+x}) [/itex]
 

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