Evaluate the limit

Question

f(x)= (1-x)/(x-2) ; lim h→0 (f(x+h)-f(x))/h

Answer 1

f(x) = (1-x)/(x-2) = (1-x)(x-2)^-1

f'(x) = (1-x)(-1)/(x-2)^2 -1/(x-2)

= (x-1)/(x-2)^2 - (x-2)/(x-2)^2

= 1/(x-2)^2= limit as h approaches zero of [f(x+h)-f(x)]/h = the derivative of f(x)

=f'(x) = denominator times derivative of numberator minus numerator times derivative of denominator, all over denominator squared = [(x-2)(-1) - (1-x)]/(x-2)^2 = [2-x +x -1] = 1/(x-2)^2

or

f(x+h) = [(1-(x+h))/(x+h-2) - (1-x)/(x-2)]/h

= [h/(x-2)(x-2+h)]/h = 1/(x-2)(x-2+h) = 1/(x-2)(x-2+0) = 1/(x-2)^2 as h approaches zero