Simplifying Limit Expressions with Rationalization: f(x) = √(x-1)

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SUMMARY

The discussion focuses on simplifying the limit expression for the function f(x) = √(x-1) using rationalization techniques. The correct approach involves rewriting the expression [f(x+h) - f(x)] / h as a single fraction and rationalizing the numerator by multiplying by the conjugate, resulting in the expression (√(x+h-1) - √(x-1)) / h. This method allows for the simplification necessary to evaluate the limit as h approaches 0, ultimately leading to the derivative of the function.

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Homework Statement



Fore each of the following functions, find the expression for [ f(x+h) - f(x)] / h. Simplify each of your expressions far enough so that plugging in h=0 would be allowed.

...

(f). f(x) = √(x-1) (Hint: Rationalize the numerator)


Homework Equations



Nothing, really.

The Attempt at a Solution




So a friend of mine asked me this question and I couldn't really figure it out. If you can, show me the simplification process.

I can get it to (x+h-1)/(h√(x+h-1)) - (x-1)/(h√(x-1))
 
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That's not how you're supposed to rationalize the numerator. Write out the limit as one fraction, and rationalize the entire numerator: sqrt(x-1+h) - sqrt(x-1). The purpose of this will become clear once you get the answer.
 

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