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

Problem

(lim_x→0)(sin(4*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, which states that (lim_θ→0)(sin(θ)/θ)=1

3. Manipulate the expression to match the identity by multiplying both the numerator and the denominator by 4

4/4⋅sin(4*x)/x

4. Rearrange the constants to isolate the limit structure.

4⋅sin(4*x)/(4*x)

5. Apply the limit property (lim_x→a)(c⋅ƒ(x))=c⋅(lim_x→a)(ƒ(x))

4⋅(lim_x→0)(sin(4*x)/(4*x))

6. Substitute u=4*x noting that as x→0 u→0

4⋅(lim_u→0)(sin(u)/u)

7. Evaluate the limit using the identity (lim_u→0)(sin(u)/u)=1

4⋅1=4

Final Answer

(lim_x→0)(sin(4*x)/x)=4

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