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

Problem

(lim_x→0)(sin2(x)/x)

Solution

1. Rewrite the expression to isolate the standard trigonometric limit.

sin2(x)/x=sin(x)⋅sin(x)/x

2. Apply the product rule for limits.

(lim_x→0)(sin(x)⋅sin(x)/x)=((lim_x→0)(sin(x)))⋅((lim_x→0)(sin(x)/x))

3. Evaluate the first limit using direct substitution.

(lim_x→0)(sin(x))=sin(0)=0

4. Evaluate the second limit using the fundamental trigonometric limit identity.

(lim_x→0)(sin(x)/x)=1

5. Multiply the results of the two limits.

0⋅1=0

Final Answer

(lim_x→0)(sin2(x)/x)=0

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