Evaluate the Limit limit as x approaches 0 of (sin(5x))/x

Problem

(lim_x→0)(sin(5*x)/x)

Solution

1. Identify the standard trigonometric limit form (lim_θ→0)(sin(θ)/θ)=1

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

(lim_x→0)((5⋅sin(5*x))/(5*x))

3. Apply the constant multiple rule for limits to move the 5 outside the limit.

5⋅(lim_x→0)(sin(5*x)/(5*x))

4. Substitute a new variable u=5*x As x→0 it follows that u→0

5⋅(lim_u→0)(sin(u)/u)

5. Evaluate the limit using the fundamental trigonometric limit identity.

5⋅1=5

Final Answer

(lim_x→0)(sin(5*x)/x)=5

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