How to Find the Limit of This Function Using Only Algebra?

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SUMMARY

The discussion focuses on finding the limit of the function lim (³√x - 4) / (√x - 8) as x approaches 64 using only algebraic manipulation, explicitly avoiding Newton's Method and L'Hôpital's Rule. The solution involves factoring using the difference of two squares and cubes, recognizing that 4 is a perfect square and 8 is a perfect cube. The approach highlights the use of x^(1/3) and x^(1/2) for the numerator and denominator, respectively, showcasing an impressive algebraic technique.

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racer
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Hello there

Find the lim of this function using Algebra alone and without using
Newton Method or L'Hapital Rule, only Algebraic manipulation is allowed.

I have previously posted this lim but no one did solve it algebriaclly, I solved it using
Algebra and uploaded the solution.

Lim [tex]\frac{^3\sqrt{x}- 4}{\sqrt{x}- 8}[/tex]
X ---> 64

The solution
http://img3.freeimagehosting.net/image.php?ae06b74212.jpg
 
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Thanks for the update, although I must have missed the original post. It is quite obvious to me that you would want to factor using the difference of two squares/cubes rule as you have 4 (a perfect square) and 8 (a perfect cube). The only thing that might make one hesitate is that instead of using x^2 and x^3 for the numerator and denominator respectively they use x^(1/3) and x(1/2).
 
That's impressive! Of course, using L'Hopital's rule (which, I notice, you are not allowed to do) would make it fairly simple.
 
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