Evaluate the Limit ( limit as x approaches 0 of sin(3x))/x

Problem

(lim_x→0)(sin(3*x)/x)

Solution

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

2. Recall the fundamental trigonometric limit identity:

(lim_θ→0)(sin(θ)/θ)=1

3. Manipulate the expression to match the identity by multiplying the numerator and denominator by 3

(lim_x→0)((3⋅sin(3*x))/(3*x))

4. Apply the constant multiple rule for limits to move the 3 outside the limit:

3⋅(lim_x→0)(sin(3*x)/(3*x))

5. Substitute u=3*x As x→0 it follows that u→0

3⋅(lim_u→0)(sin(u)/u)

6. Evaluate the limit using the identity, which gives 3⋅1=3

Final Answer

(lim_x→0)(sin(3*x)/x)=3

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