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

In summary, for the function f(x) = √(x-1), the expression for [ f(x+h) - f(x)] / h is (sqrt(x-1+h) - sqrt(x-1)) / (h√(x+h-1)). This expression has been simplified to allow for plugging in h=0. The process to simplify the expression involves rationalizing the entire numerator by writing the limit as one fraction.
  • #1
Jamin2112
986
12

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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  • #2
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.
 

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

What is rationalization?

Rationalization is the process of simplifying an expression by removing any radicals from the denominator of a fraction.

Why is it important to rationalize limit expressions?

Rationalizing limit expressions allows us to evaluate them more easily, as it eliminates any irrational numbers and simplifies the expression.

How do you rationalize a limit expression with a square root?

To rationalize a limit expression with a square root, we multiply the numerator and denominator by the conjugate of the denominator, which is the same expression but with the opposite sign between the terms in the middle.

Can we rationalize any limit expression?

Yes, we can rationalize any limit expression with a square root in the denominator by using the conjugate method.

What is the resulting expression after rationalizing the given limit expression?

The resulting expression after rationalizing the given limit expression, f(x) = √(x-1), is (x-1)/(√(x-1)).

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