Evaluate the Limit

Problem

(lim_x→0)(sin(3*x)/(2*x))

Solution

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

2. Apply the fundamental trigonometric limit property, which states that (lim_θ→0)(sin(θ)/θ)=1

3. Rewrite the expression to match the property by factoring out the constant 1/2 from the denominator.

1/2*(lim_x→0)(sin(3*x)/x)

4. Manipulate the fraction to create the argument 3*x in the denominator by multiplying both the numerator and denominator by 3

3/2*(lim_x→0)(sin(3*x)/(3*x))

5. Substitute the known limit value (lim_3*x→0)(sin(3*x)/(3*x))=1 into the expression.

3/2⋅1=3/2

Final Answer

(lim_x→0)(sin(3*x)/(2*x))=3/2

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