Need help dealing w/ limits!?

Question

For which of the following functions f and corresponding #s a is the limit the value of f' (a)?
lim (1+h)^8-1 / h
h->0

A) f(x)= x^8, a =0
B) f(x)= (x-1)^8, a =1
C) f(x)=(x+1) ^8, a =8
D) f(x)= x^8, a=1
E) f(x)= (x+1)^8, a =1
F) f(x)= x^8, a=8

Answer 1

lim [f(a+h) - f(a)] / h = f'(a)
h->0

consider 2 terms: f(a+h) and f(a).

f(a+h):
Here, f(a+h) = (1+h)^8 so a = 1
why?
because on left hand side 'a' is constant and 'h' is variable and on right side 1 is constant and 'h' is variable

so we have to select from choices B), D) & E)

next, here f(a) = 1
for B), f(a) = f(1) = (1-1)^8 = 0 [f(a) = 1 not satisfied]
for D), f(a) = f(1) = (1)^8 = 1 [f(a) = 1 satisfied]
for E), f(a) = f(1) = (1+1)^8 = 2^8 [f(a) = 1 not satisfied]

Thus D) is correct answer.

verify:
lim [f(a+h) - f(a)] / h = f'(a)
h->0
when a = 1 from D) f(x) = x^8
so f'(a) =
lim [f(1+h) - f(1)] / h
h->0
= lim [(1+h)^8 - 1^8] / h
h->0
= lim [(1+h)^8 - 1] / h
h->0

Answer 2

((1 + h)^8 - 1)/h =
h^7 + 8h^6 + 28h^5 + 56h^4 + 98h^3 + 56h^2 + 28h + 8
limit = 8
h->0
C) f(x)=(x+1) ^8, a =8

Answer 3

you need more info, in order to have a limit on f'(a) it has to be approaching something... why do you have a "lim f(h) " below the problem?