Evaluate the Limit limit as x approaches 0 of (sin(9x))/x

Problem

(lim_x→0)(sin(9*x)/x)

Solution

1. Identify the fundamental trigonometric limit formula, which states that for any variable u

(lim_u→0)(sin(u)/u)=1

2. Manipulate the expression to match the form of the fundamental limit by multiplying the numerator and denominator by 9

(lim_x→0)((9⋅sin(9*x))/(9*x))

3. Apply the constant multiple rule for limits to move the constant 9 outside the limit:

9⋅(lim_x→0)(sin(9*x)/(9*x))

4. Substitute u=9*x As x→0 it follows that u→0

9⋅(lim_u→0)(sin(u)/u)

5. Evaluate the limit using the fundamental formula:

9⋅1=9

Final Answer

(lim_x→0)(sin(9*x)/x)=9

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