lim x->0 (Sin^3 3x - 3x^3)/(x^3) (x^4)

The left term can be seen as

(sin3x/x)^3 - 3
= 27(sin3x/3x)^3 - 3

now, we all know that lim sinu/u = 1 as u->0, so the limit of all that is just

27-3 = 24

multiply that by x^4 and you just wind up with 0 as x->0

Still not sure what that x^4 is doing in there. It just clutters up the concept of the 0/0 limit on the left.

To solve this limit, we can simplify the given expression. Let's break it down step by step.

First, let's look at the numerator: sin^3(3x) - 3x^3. We can apply the difference of cubes formula to rewrite 3x^3 as (sin(3x))^3 - (3x)^3.

Next, let's focus on the denominator: x^3 * (x^4). We can simplify this by multiplying the x^3 and x^4 terms together, resulting in x^7.

Now, let's rewrite the entire expression: [(sin(3x))^3 - (3x)^3]/(x^3 * x^4), which is equal to [(sin(3x))^3 - (3x)^3]/(x^7).

Now, we can factor out a common factor of (sin(3x))^3 - (3x)^3 in the numerator. This gives us [(sin(3x))^3 - (3x)^3] * (1/x^7).

Now that we have factored out the common factor, we can simplify further. As x approaches 0, the term 1/x^7 goes to infinity, so our limit simplifies to:

lim x->0 [(sin(3x))^3 - (3x)^3] * (1/x^7)

Now, we can substitute x = 0 into the expression:

[(sin(3(0)))^3 - (3(0))^3] * (1/(0)^7
= (sin(0))^3 - 0 * (1/0^7)
= 0^3 - 0 * (1/0^7)
= 0 - 0 * (1/0^7)
= 0 * (1/0^7)
= 0.

Therefore, the limit of the given expression as x approaches 0 is 0.