Limit of \( \displaystyle \frac{\sin^{2}{\left(3 x \right)}}{9 x^{2}} \) as \( x \to 0 \)

Problem 1.171 · medium

1. \[ \lim_{x \to 0^+}\left(\frac{\sin^{2}{\left(3 x \right)}}{9 x^{2}}\right) \]
   limit algebraStart with the original limit. Rewrite the expression as a square.✓ Proved
2. \[ = \left(\lim_{x \to 0^+}\left(\frac{\sin{\left(3 x \right)}}{3 x}\right)\right)^{2} \]
   limit-lawApply the power rule for limits.✓ Proved
3. \[ = \left(\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(3 x \right)}}{\frac{d}{d x} 3 x}\right)\right)^{2} \]
   lhopitalApply L'Hopital's rule to the inner limit.✓ Proved
4. \[ = \left(\lim_{x \to 0^+} \cos{\left(3 x \right)}\right)^{2} \]
   simplify simplifyCompute the derivatives. Simplify the fraction inside the limit.✓ Proved
5. \[ = 1 \]
   limit simplifyEvaluate the limit of the cosine function. Final result.✓ Proved