find the limit if f((8+h)-f(8))/h .if f(x)= (472-2 x^2)^1/3.
THE LIMIT DOES NOT EXIST!! (MEAN GIRLS REFERENCE) LOL.
NOT THE ANSWER THO!!
h=0 but i don't know how to solve those kind of limits.
f(x)=(472 - 2x^2)^1/3
so,
f(8) = (472 - 128)^1/3 = 344^1/3
f(8+h) = (472 - 2(8+h)^2)^1/3
= (344 - (32h+2h^2))^1/3
By the binomial theorem, that is
344^1/3 - (1/3)(344^-2/3)(32h+2h^2) + a bunch of stuff with higher powers of h
Now subtract and you have f((8+h)-f(8))
= (-1/3)(344^-2/3)(2h(16+h^2))
Now divide that by h and you are left with
(-2/3)(344^-2/3)(16+h^2)
The limit as h->0 is thus
(-32/3)(344^-2/3)
To find the limit of the given expression, we first need to substitute the given function f(x) into the expression, and then evaluate the limit as h approaches 0.
Given function: f(x) = (472 - 2x^2)^(1/3)
Let's substitute f(x) into the expression:
f((8 + h) - f(8))/h
= (f(8 + h) - f(8))/h
= ((472 - 2(8 + h)^2)^(1/3) - (472 - 2(8)^2)^(1/3))/h
Now we can calculate the limit by evaluating this expression as h approaches 0.
lim(h->0) ((472 - 2(8 + h)^2)^(1/3) - (472 - 2(8)^2)^(1/3))/h
To simplify the calculation, we can start by expanding and simplifying the terms inside the parentheses:
lim(h->0) ((472 - 2(64 + 16h + h^2))^(1/3) - (472 - 2(64))^(1/3))/h
= lim(h->0) ((472 - 2(64 + 16h + h^2))^(1/3) - (472 - 128)^(1/3))/h
= lim(h->0) ((472 - 128 - 32h - 2h^2)^(1/3) - (472 - 128)^(1/3))/h
= lim(h->0) ((344 - 32h - 2h^2)^(1/3) - 344^(1/3))/h
Now, we can simplify this expression further by using the limit laws of algebra. We can divide each term by h separately:
lim(h->0) (344^(1/3))/h + lim(h->0) ((-32h)/h) - lim(h->0) ((2h^2)/h))/(h)
= (344^(1/3))/lim(h->0) h + lim(h->0) (-32) - lim(h->0) (2h))/lim(h->0) h
Since the limit of h as it approaches 0 is 0, we have:
(344^(1/3))/0 + lim(h->0) (-32) - lim(h->0) (2h))/0
= ∞ - 32 - 0
= ∞ - 32
= ∞
Therefore, the limit of f((8 + h) - f(8))/h as h approaches 0 is ∞.