the limit (h->0) of
(sin^2(3x + 3h) - sin^2(3x))/(h)
is....
To find the limit as h approaches 0 of (sin^2(3x + 3h) - sin^2(3x))/(h), we can use the limit definition of the derivative. Let's break down the steps to find the limit:
Step 1: Use the identity sin^2(A) = (1/2)(1 - cos(2A)) to simplify the expression.
(sin^2(3x + 3h) - sin^2(3x))/(h)
= [(1/2)(1 - cos(2(3x + 3h)))] - [(1/2)(1 - cos(2(3x)))] / h
Step 2: Expand the cos(2(3x + 3h)) and cos(2(3x)) terms.
[(1/2)(1 - cos(6x + 6h))] - [(1/2)(1 - cos(6x))] / h
Step 3: Distribute the 1/2 factor to simplify the expression further.
[1/2 - 1/2cos(6x + 6h)] - [1/2 - 1/2cos(6x)] / h
Step 4: Combine like terms within each of the numerator terms.
[1/2 - 1/2cos(6x)cos(6h) + 1/2sin(6x)sin(6h)] - [1/2 - 1/2cos(6x)] / h
Step 5: Cancel out the common terms in each of the numerator terms.
- 1/2cos(6x)cos(6h) + 1/2sin(6x)sin(6h) / h
Step 6: Factor out a cos(6x) from the numerator.
cos(6x)(- 1/2cos(6h)) + 1/2sin(6x)sin(6h) / h
Now, we can find the limit as h approaches 0. Let's evaluate each term:
lim(h->0) [cos(6x)(- 1/2cos(6h)) + 1/2sin(6x)sin(6h)] / h
Since h is approaching 0, we can substitute 0 for h in each term:
= cos(6x)(- 1/2cos(0)) + 1/2sin(6x)sin(0) / 0
= cos(6x)(- 1/2) + 1/2sin(6x)(0) / 0
= - 1/2cos(6x) + 0 / 0
Now, we have an indeterminate form of 0/0. This means we need to further simplify the expression. To do this, we can differentiate the numerator and denominator separately and find the limit again. However, we cannot solve this further without additional information or restrictions on the function.
Therefore, the limit as h approaches 0 of (sin^2(3x + 3h) - sin^2(3x))/(h) is indeterminate (0/0).