limx-->3 sin 3x/4x

To find the limit of a function as x approaches a certain value, you can apply direct substitution by plugging in the value that x is approaching into the function. However, in this case, directly substituting x = 3 into the function would result in an indeterminate form of 0/0.

To evaluate a limit like this, you can use L'Hôpital's Rule, which states that if the limit of the ratio of two functions f(x)/g(x) is of the indeterminate form 0/0 or ∞/∞, then the limit of the ratio can be obtained by evaluating the limit of the derivatives of the numerator and denominator.

In order to apply L'Hôpital's Rule, you need to take the derivative of the numerator and denominator separately and find their limits. Let's start by finding the derivative of the numerator, which is sin(3x):

f'(x) = 3cos(3x)

Next, let's find the derivative of the denominator, which is 4x:

g'(x) = 4

Now, we can rewrite the original limit as the limit of the ratio of the derivatives:

lim(x→3) [3cos(3x)] / [4]

Since x is still approaching 3, we can evaluate this limit directly:

lim(x→3) [3cos(3x)] / [4] = [3cos(9)] / [4]

Therefore, the limit of (sin 3x) / (4x) as x approaches 3 is [3cos(9)] / [4].