Evaluate the Limit

Problem

(lim_x→0)(sin(4*x)/sin(2*x))

Solution

1. Identify the indeterminate form by substituting x=0 into the expression, which results in sin(0)/sin(0)=0/0

2. Apply the double-angle identity for sine, sin(2*θ)=2*sin(θ)*cos(θ) to the numerator by letting θ=2*x

3. Substitute the identity into the limit expression.

(lim_x→0)((2*sin(2*x)*cos(2*x))/sin(2*x))

4. Cancel the common factor of sin(2*x) from the numerator and the denominator, provided x≠0

(lim_x→0)(2)*cos(2*x)

5. Evaluate the limit by substituting x=0 into the remaining expression.

2*cos(2*(0))=2*cos(0)

6. Simplify the result using the fact that cos(0)=1

2*(1)=2

Final Answer

(lim_x→0)(sin(4*x)/sin(2*x))=2

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