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