help me please.?

Question

Evaluate lim (3 + h)-1 - 3-1 / h
h>0

Written: Evaluate limit h apporaching 0. (3 + h) to the negative 1 power - 3 to the negative 1 power all over h.

Answer 1

Well, I guess you mean lim h -> 0 ((3 + h)^(-1) - 3^(-1))/h. We know that, when h -> 0, then (1 + h)(-1) ~ 1 - h.

So, we have h -> 0 ((3 + h)^(-1) - 3^(-1))/h.= lim h -> 0 (3^(-1)(1 + h/3)^(-1) - 3^(-1))/h = lim h -> 0 3^(-1) [1 - h/3 -1]/h = lim h-> 0 3^(-1) -(h/3)h = -3^(-1) * 1/3 = -1/9.

If you have already studied derivatives, you can notice this is just the derivative of f(x) =1/x = x^(-1) at x = 3. We knaow that f'(x) = -1/x^2, so if x = 3, then f'(x) = f'(3) = -1/9.