Evaluate the Limit limit as x approaches 0 of (sin(4x))/x

Problem

(lim_x→0)(sin(4*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 4

(lim_x→0)((4⋅sin(4*x))/(4*x))

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

4⋅(lim_x→0)(sin(4*x)/(4*x))

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

4⋅(lim_u→0)(sin(u)/u)

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

4⋅1=4

Final Answer

(lim_x→0)(sin(4*x)/x)=4

Want more problems? Check here!