Evaluate the Limit limit as x approaches 0 of (sin(2x))/x

Problem

(lim_x→0)(sin(2*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 2

(lim_x→0)((2*sin(2*x))/(2*x))

4. Apply the constant multiple rule for limits to move the factor of 2 outside the limit.

2⋅(lim_x→0)(sin(2*x)/(2*x))

5. Substitute u=2*x noting that as x→0 u→0

2⋅(lim_u→0)(sin(u)/u)

6. Evaluate the limit using the identity.

2⋅1=2

Final Answer

(lim_x→0)(sin(2*x)/x)=2

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