Find lim h -> 0 [f(x+h) - f(x)]/h for f(x) = sqr(x-8)

Question

options

a. 1/[8 sqr(x-8)]

b. 1/[2 sqr(x-8)]

c. [sqr(x-a)/2

d. sqr(x-8)

e. limit does not exist.

Answer 1

[f(x+h) - f(x)] / h

= [sqr(x+h-8) -sqr(x-8)]/h

To solve this, you should multiply both numerator and denominator by the conjugate of the numerator. It seems complicated but it will let you simplify eventually.

[sqr(x+h-8) - sqr(x-8)] * [sqr(x+h-8) + sqr(x-8)] ÷ [ h *[sqr(x+h-8) + sqr(x-8)]]

Now foil the numerator and you should get:

[(x +h -8) - (x-8)] ÷ [ h *[sqr(x+h-8) + sqr(x-8)]]

simplifying gives you h ÷ [ h *[sqr(x+h-8) + sqr(x-8)]]

h cancels, so you're left with 1 ÷ [sqr(x+h-8) + sqr(x-8)] = 1 ÷ 2sqr(x-8) when you plug in h=0