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

• Jamin2112
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.
Jamin2112

## 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)

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))

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.

## 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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