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 sin(0)sin(0)−sin(0)=00 which is an indeterminate form. Therefore, we can apply L'H o^ pital's Rule, which states that if limx→cg(x)f(x) is of the form 00or∞∞, then limx→cg(x)f(x)=limx→cg′(x)f′(x).
Step 2: Differentiate numerator and denominator
We differentiate the numerator and the denominator with respect to x The derivative of sin(ax)isacos(ax), so dxd(sin(5x))=5cos(5x) and dxd(sin(3x))=3cos(3x) The derivative of sin(x)iscos(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 15(1)−3(1) = 15−3 = 2$. This is the final value of the limit.