Question

Question #d61d7

Answer 1

`lim_(h->0)(f(x+h)-f(x))/h=2x-7`

Explanation:

We know what `f(x)` is, `f(x+h)` is just `f(x)` with `x+h` instead of `x`

`f(x+h)=(x+h)^2-7(x+h)+3`
`f(x)=x^2-7x+3`

`lim_(h->0)(f(x+h)-f(x))/h=lim_(h->0)((x+h)^2-7(x+h)+3-(x^2-7x+3))/h`
`color(white)(lim_(h->0)(f(x+h)-f(x))/h)=lim_(h->0)(x^2+2xh+h^2-7x-7h+3-x^2+7x-3)/h`
`color(white)(lim_(h->0)(f(x+h)-f(x))/h)=lim_(h->0)(2xh+h^2-7h)/h`
`color(white)(lim_(h->0)(f(x+h)-f(x))/h)=lim_(h->0)2x+h-7`
`color(white)(lim_(h->0)(f(x+h)-f(x))/h)=lim_(h->0)2x+0-7`
`color(white)(lim_(h->0)(f(x+h)-f(x))/h)=2x-7`