Limit of \( \displaystyle \frac{\sin^{2}{\left(3 x - 1 \right)}}{\left(3 x - 1\right)^{2}} \) as \( x \to \frac{1}{3} \)

Problem 1.422 · medium

1. \[ \lim_{x \to \frac{1}{3}^+}\left(\frac{\sin^{2}{\left(3 x - 1 \right)}}{\left(3 x - 1\right)^{2}}\right) \]
   limit algebra algebraStart with the original limit. Rewrite the square of the fraction. Group the terms to reveal the structure.✓ Proved
2. \[ = \lim_{x \to \frac{1}{3}^+}\left(\frac{\sin^{2}{\left(3 x - 1 \right)}}{9 \left(x - \frac{1}{3}\right)^{2}}\right) \]
   algebraFactor out the constant 3 from the denominator.✓ Proved
3. \[ = \lim_{u \to 0^+}\left(\frac{\sin^{2}{\left(3 u \right)}}{9 u^{2}}\right) \]
   substitution simplifyLet u = x - 1/3, so u approaches 0. Simplify the expression inside the limit.✓ Proved
4. \[ = \lim_{u \to 0^+} 1 \]
   trig-limit simplifyUse the fundamental trigonometric limit sin(u)/u -> 1. Evaluate the constant expression.✓ Proved
5. \[ = 1 \]
   limitThe limit evaluates to 1.✓ Proved