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

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The discussion focuses on finding the limit of the function (3√x - 4) / (√x - 8) as x approaches 64 using only algebraic manipulation. The original poster emphasizes that previous attempts to solve this limit algebraically were unsuccessful, and they have since provided their own solution. Participants suggest factoring techniques, particularly using the difference of squares and cubes, to simplify the expression. They note the challenge posed by the different powers in the numerator and denominator, but acknowledge that algebraic methods can yield a solution without resorting to L'Hôpital's rule. The conversation highlights the importance of algebraic skills in evaluating limits.
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 \frac{^3\sqrt{x}- 4}{\sqrt{x}- 8}
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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