Evaluate the Limit

Problem

(lim_x→0)(sin(4*x)/sin(6*x))

Solution

1. Identify the indeterminate form by substituting x=0 into the expression, which results in sin(0)/sin(0)=0/0

2. Apply L'Hôpital's Rule, which states that if a limit results in 0/0 the limit is equal to the limit of the derivatives of the numerator and denominator.

3. Differentiate the numerator sin(4*x) using the chain rule to get 4*cos(4*x)

4. Differentiate the denominator sin(6*x) using the chain rule to get 6*cos(6*x)

5. Substitute x=0 into the new expression (4*cos(4*x))/(6*cos(6*x))

6. Simplify the result using the fact that cos(0)=1

Final Answer

(lim_x→0)(sin(4*x)/sin(6*x))=2/3

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