Evaluate the Limit

Problem

(lim_x→0)(sin2(3*x)/(x2))

Solution

1. Rewrite the expression to group the squared terms together using the property (a2)/(b2)=(a/b)2

(lim_x→0)(sin(3*x)/x)

2. Apply the limit power rule, which allows the limit to be moved inside the square.

((lim_x→0)(sin(3*x)/x))2

3. Manipulate the fraction to match the standard limit form (lim_u→0)(sin(u)/u)=1 by multiplying the numerator and denominator by 3

((lim_x→0)((3*sin(3*x))/(3*x)))2

4. Factor out the constant 3 from the limit.

(3⋅(lim_x→0)(sin(3*x)/(3*x)))2

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

(3⋅1)2

6. Simplify the resulting numerical expression.

3=9

Final Answer

(lim_x→0)(sin2(3*x)/(x2))=9

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