f(x)=x/x-2, (3,3)

Question

I need to find the slope of the function, then find the equation for the line tangent

with lim h->0 f(x+h)-f(x)/h

Answer 1

f(x)=x/(x-2), f(x+h)=(x+h)/(x+h-2).

(f(x+h)-f(x))/h=

((x+h)/(x+h-2)-x/(x-2))/h=((x+h)(x-2)-x(x+h-2))/(h(x-2+h)(x-2)) for general x,

(f(x+h)-f(x))/h=(x²-2x+hx-2h-x²-hx+2x)/(h(x²-4x+4+hx-2h))=

-2h//(h(x²-4x+4+hx-2h))=-2/(x²-4x+4+hx-2h).

As h→0 this becomes -2/(x²-4x+4) or -2/(x-2)². This is the derivative of f(x).

At the point (3,3) x=3 so the slope is -2/(3-2)²=-2.