Find the limit as h approaches 0 of sin²(3x+3h) - sin²(3x) / h
To find the limit as h approaches 0 of the given expression, we can use limits and trigonometric identities.
Let's simplify the given expression first:
sin²(3x+3h) - sin²(3x) / h
Expanding the terms using the sum-to-product trigonometric identity (sin(A+B) = sin(A)cos(B) + cos(A)sin(B)) in the first part of the expression:
(sin(3x)cos(3h) + cos(3x)sin(3h))² - sin²(3x) / h
Now, we can apply the identity (a² - b² = (a - b)(a + b)) to simplify the numerator:
((sin(3x)cos(3h) + cos(3x)sin(3h)) - sin(3x))((sin(3x)cos(3h) + cos(3x)sin(3h)) + sin(3x)) / h
Next, we can cancel out the common factor of sin(3x) in both terms:
(cos(3h)sin(3h))((sin(3x)cos(3h) + cos(3x)sin(3h)) + sin(3x)) / h
Now, we can simplify the expression further by factoring out sin(3h) as a common factor:
sin(3h)(cos(3h)(sin(3x)cos(3h) + cos(3x)sin(3h)) + sin(3x)) / h
Using the trigonometric identity (sin(2θ) = 2sin(θ)cos(θ)) to simplify sin(3x)cos(3h) + cos(3x)sin(3h):
sin(3h)(cos(3h)(2sin(3x)cos(3x)) + sin(3x)) / h
Next, we simplify further:
2sin(3h)(cos²(3x) - sin²(3x)) + sin(3x)cos(3h) / h
Using the identity (cos²(θ) = 1 - sin²(θ)):
2sin(3h)((1 - sin²(3x)) - sin²(3x)) + sin(3x)cos(3h) / h
Expanding and simplifying:
2sin(3h)(1 - 2sin²(3x)) + sin(3x)cos(3h) / h
Now, let's find the limit as h approaches 0:
lim(h->0) 2sin(3h)(1 - 2sin²(3x)) + sin(3x)cos(3h) / h
To evaluate this limit, we can break it into two separate limits:
lim(h->0) 2sin(3h)(1 - 2sin²(3x)) / h + lim(h->0) sin(3x)cos(3h) / h
For the first limit, we can use the standard limit formula:
lim(h->0) sin(θ) / θ = 1
Applying this, our first limit becomes:
2(1 - 2sin²(3x)) + lim(h->0) sin(3x)cos(3h) / h
In the second limit, we can use another trigonometric limit:
lim(h->0) sin(θ) / h = θ
Applying this, our second limit becomes:
2(1 - 2sin²(3x)) + sin(3x)cos(0)
Since cos(0) = 1, our second limit simplifies to:
2(1 - 2sin²(3x)) + sin(3x)
Finally, we can simplify the expression further:
2 - 4sin²(3x) + sin(3x)
This is the simplified expression for the given limit as h approaches 0.