Evaluate the Limit

Problem

(lim_x→0)(sin(5*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 isolate the constant factor in the denominator by moving the 3 outside the limit.

1/3*(lim_x→0)(sin(5*x)/x)

4. Manipulate the expression to match the argument of the sine function by multiplying the numerator and denominator by 5

5/3*(lim_x→0)(sin(5*x)/(5*x))

5. Substitute the known limit value (lim_5*x→0)(sin(5*x)/(5*x))=1 into the expression.

5/3⋅1

Final Answer

(lim_x→0)(sin(5*x)/(3*x))=5/3

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