Let``` ` `f(x)=x^2-2x+8`, what is `lim_(h->0) (f(x+h)-f(x))/h`

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For any function f(x), the limit `lim_(h->0)(f(x+h)-f(x))/h` is the derivative f'(x).

As f(x) = x^2 - 2x + 8, f'(x) = 2x - 2

**For the function `f(x) = x^2 - 2x + 8` , **`lim_(h->0)(f(x+h)-f(x))/h = 2x - 2`

The above response is correct. `lim_(h->0) (f(x+h)-f(x))/(h)` is the derivative f'(x), but if the function f(x) needs to be evaluated using the limit equation, this is the method to approach the problem.

The limit function tells us that we first need to substitute (x+h) for "x" in f(x), and then subtract it with the actual f(x). Therefore:

`lim_(h->0) (f(x+h)-f(x))/(h)` ` ``= lim_(h->0) ((x+h)^2-2(x+h)+8-(x^2-2x+8))/h`

`=> lim_(h->0) (x^2+2xh+h^2-2x-2h+8-x^2+2x-8)/h`

After simplifying the fraction, we can take the limit by substituting 0 for h.

`=> lim_(h->0) (2x+h-2)`

`=> 2x+0-2 = 2x-2`

Therefore, the answer is **2x-2.**

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