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Finding the Limit of a Rational Function

  • Thread starter Manarius
  • Start date
  • #1
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This is just a problem I came across while reviewing basic calculus.

Homework Statement



Find the limit as x approaches 0 of f(x)=(1/(x(x+1)^1/2)) - (1/x)

Homework Equations





The Attempt at a Solution



My problem here is really more of an algebra problem than a calculus problem. I cannot for the life of me remember how to get the limit of the denominator of either term to not equal 0. I've tried everything I can think of (rationalizing the denominator, etc.), to no avail.

I could find no tutorials either online or in my book that use an example quite like this one. It's truly maddening, especially because I know I should have learned this 6 years ago in Algebra.

Thanks in advance. Hope you can help.
 

Answers and Replies

  • #2
ehild
Homework Helper
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1,876
Do you mean

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

?

If yes, factor out 1/x and multiply and divide f(x) by [1/sqrt(x+1)+1].

ehild
 
  • #3
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I think I must be misunderstanding what you're saying. Either that or I did the math wrong multiple times.

Would you mind clarifying what exactly you mean? Thanks.
 
  • #4
ehild
Homework Helper
15,494
1,876
Use the notation 1/√(x+1)=a.

f(x)=(1/x) (a-1)

Multiply and divide by (a+1)

f(x)=(1/x)[(a-1)(a+1)]/(a+1)

f(x)=(1/x)(a2-1)/(a+1).

Replace back 1/√(x+1) for a and simplify. Find the limit.
 

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