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Each limit represents the derivative of some function $ f $ at some number $ a $. State such an $ f $ and $ a $ in each case.

$ \displaystyle \lim_{h \to 0} \frac{\sqrt{9 + h} - 3}{h} $

$f(x)=\sqrt{x}$

$a=9$

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if f of x is equal to the square root of X. And if A Is the number nine than using the definition to limit definition of the derivative, F prime at A would be Let me raise that. I got to put into words, limit F prime of A would be to limit as H approaches zero of the function evaluated at X plus h minus two function evaluated at X uh over each. Uh And actually instead of X since we're evaluating the derivative at A instead of X, we're going to put a. So F prime of A will be limited as H approaches zero of the function at a plus H minus two function of F at a over H. No, using uh this definition for the function F of X is the square root of X. And our number A is nine. I'll continue down here. Uh This will equal the limit As a church approaches zero. Now the function F of a plus H is going to be the square root of a plus H minus the function at a f of X is squared of X. So if a baby is a square root of X, all of this is divided by H. And we're taking the limit of this expression as H goes towards zero. That's how we're gonna find uh F prime of X. Keep in mind, let's scroll up just a minute. So you remember remember that in the very top of the problem here. Yeah. Uh F of X. I defined to be the function squared of X. And a the number A equals nine. So let's go ahead and plug into number nine. Since that is um our actual uh what we're looking for, So f prime of nine, The derivative of our function squared of x at a equals nine is going to be to limit As a church approaches zero. Now the square root of A plus H is the square root of nine plus H. Since a. is nine, subtract the square root of A. Since A is nine, the square root of nine is three. And this is being divided by each. And so you can see uh the derivative F prime at nine equals to limit of the square root of nine plus H minus three over H as H approaches zero. When remember what our function And what are # eight was? This is true when F of X is equal to square root of X. And our number A. Was nine.