Given f(x) = x3, evaluate lim h->o f(a+h)-f(a)/h
f(a+h)= (a+h)^3= (a^2+ah+h^2)(a+h)=a^3+ha^2+h^2a+ha^2+ah^2+h^3
f(a)= a^3
so f(a+h)-f(a)= a^3+ha^2+h^2a+ha^2+ah^2+h^3-a^3
= ha^2+h^2a+ha^2+ah^2+h^3
dividing that by h gives
= a^2 + ah+ a^2 +ah^2 + h^2
= 2a^2 as the h approaches zero
I respectfully have to disagree with bob
(x+h)^3 = a^3 + 3a^2h + 3ah^2 + h^3
which when inserted in the above calculation results in a final of
3a^2
To evaluate the limit of the expression (f(a+h) - f(a)) / h as h approaches 0, we can follow these steps:
Step 1: Substitute the given function f(x) = x^3 into the expression.
(f(a+h) - f(a)) / h = ((a + h)^3 - a^3) / h
Step 2: Expand the numerator using the binomial expansion formula (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3.
((a + h)^3 - a^3) / h = (a^3 + 3a^2h + 3ah^2 + h^3 - a^3) / h
Step 3: Simplify the numerator by canceling out the a^3 terms.
((a + h)^3 - a^3) / h = (3a^2h + 3ah^2 + h^3) / h
Step 4: Factor out an h in the numerator.
(3a^2h + 3ah^2 + h^3) / h = h(3a^2 + 3ah + h^2) / h
Step 5: Cancel out the h in the numerator and denominator.
h(3a^2 + 3ah + h^2) / h = 3a^2 + 3ah + h^2
The expression 3a^2 + 3ah + h^2 is the simplified form of (f(a+h) - f(a)) / h. This does not depend on h and does not have an indeterminate form, so we can evaluate the limit as h approaches 0 by directly substituting h = 0 into the expression.
lim h->0 (3a^2 + 3ah + h^2) = 3a^2 + 0 + 0 = 3a^2
Therefore, the limit of (f(a+h) - f(a)) / h as h approaches 0 is 3a^2.