Solution:

To evaluate the limit of the given function as x tends to 0, we can use L'Hôpital's Rule, which states that if the limit of a function f(x)/g(x) as x approaches a is of the form 0/0 or ∞/∞, then the limit of the ratio is equal to the limit of the ratio of the derivatives of f(x) and g(x) as x approaches a.

Let's apply L'Hôpital's Rule to the given function:

Step 1: Rewrite the function in a form suitable for applying L'Hôpital's Rule.

lim(x → 0) (1 - cos(3x))/(x^2)

Step 2: Differentiate the numerator and the denominator.

The derivative of 1 - cos(3x) with respect to x is 3sin(3x).
The derivative of x^2 with respect to x is 2x.

Step 3: Rewrite the limit using the derivatives.

lim(x → 0) (3sin(3x))/(2x)

Step 4: Now, let's evaluate the limit.

We can directly substitute x = 0 into the expression to get:

lim(x → 0) (3sin(3x))/(2x) = (3sin(0))/(2(0)) = 0/0

Since we still have an indeterminate form of 0/0, we can apply L'Hôpital's Rule again.

Step 5: Differentiate the numerator and the denominator.

The derivative of 3sin(3x) with respect to x is 9cos(3x).
The derivative of 2x with respect to x is 2.

Step 6: Rewrite the limit using the derivatives.

lim(x → 0) (9cos(3x))/(2)

Step 7: Now, let's evaluate the limit.

We can substitute x = 0 into the expression to get:

lim(x → 0) (9cos(3x))/(2) = (9cos(0))/(2) = 9/2

Therefore, the limit of (1 - cos(3x))/(x^2) as x tends to 0 is 9/2.