use L'Hopital's Rule to evaluate lim (4x(cos 8x-1))/(sin 8x - 8x) as x->0

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To use L'Hopital's Rule, we start by finding the derivatives of the numerator and denominator separately.

Let's begin by finding the derivative of the numerator.

Take the derivative of 4x (which is a simple power rule):
f'(x) = 4

To find the derivative of (cos 8x - 1), we use the chain rule:
f'(x) = -sin(8x) * 8

Now, let's move on to finding the derivative of the denominator.

The derivative of sin x is simply cos x:
g'(x) = cos(8x)

Now that we have the derivatives of the numerator and denominator, we can evaluate the limit.

The limit of the function as x approaches 0 is given by:
lim x->0 (4x(cos 8x - 1))/(sin 8x - 8x)

Using L'Hopital's Rule, we divide the derivatives:
lim x->0 (4)/((cos 8x) * 8)

Now we can substitute x = 0 into the expression:
lim x->0 (4)/(cos 8x * 8)

Since cos(0) = 1, we have:
lim x->0 (4)/(1 * 8)

Simplifying further:
lim x->0 (1)/(2)

Therefore, the limit as x approaches 0 is 1/2.