Evaluate the Limit

Problem

(lim_x→0)(sin(3*x)/(5*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 multiplying the numerator and denominator by 3 to create the 3*x term in the denominator.

(lim_x→0)(sin(3*x)/(5*x))=(lim_x→0)(3/5)⋅sin(3*x)/(3*x)

4. Factor out the constant 3/5 from the limit.

3/5⋅(lim_x→0)(sin(3*x)/(3*x))

5. Evaluate the limit using the property (lim_3*x→0)(sin(3*x)/(3*x))=1

3/5⋅1=3/5

Final Answer

(lim_x→0)(sin(3*x)/(5*x))=3/5

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