Evaluate: lim(x->0) (sin(5x) - sin(3x)) / (sin(x)).

Step 1: Apply L'Hôpital's Rule

First, we check the limit by direct substitution. Substituting $x=0$ into the expression gives $\frac{\sin(0) - \sin(0)}{\sin(0)} = \frac{0}{0}$, which is an indeterminate form. Therefore, we can apply L'Hôpital's Rule, which states that if $\lim_{x\to c} \frac{f(x)}{g(x)}$ is of the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then $\lim_{x\to c} \frac{f(x)}{g(x)} = \lim_{x\to c} \frac{f'(x)}{g'(x)}$.

Step 2: Differentiate numerator and denominator

We differentiate the numerator and the denominator with respect to x. The derivative of $\sin(ax)$ is a $\cos(ax)$, so $\frac{d}{dx}(\sin(5x)) = 5\cos(5x)$ and $\frac{d}{dx}(\sin(3x)) = 3\cos(3x)$. The derivative of $\sin(x)$ is $\cos(x)$.

Step 3: Substitute $x=0$

Now that we have differentiated the numerator and the denominator, we can substitute $x=0$ into the new expression. We know that $\cos(0) = 1$.

Step 4: Calculate the final value

Substituting $\cos(0) = 1$ into the expression, we get $\frac{5(1) - 3(1)}{1}$ = $\frac{5 - 3}{1}$ = 2. This is the final value of the limit.

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