Evaluate the Limit

Problem

(lim_x→0)(sin(2*x)/(3*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 argument of the sine function by multiplying the numerator and denominator by 2/2

(lim_x→0)(sin(2*x)/(3*x))=(lim_x→0)((2*sin(2*x))/(2⋅3*x))

4. Factor out the constant coefficients from the limit.

2/3⋅(lim_x→0)(sin(2*x)/(2*x))

5. Substitute the known limit value where (lim_2*x→0)(sin(2*x)/(2*x))=1

2/3⋅1=2/3

Final Answer

(lim_x→0)(sin(2*x)/(3*x))=2/3

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