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

Problem 1.411 · medium

1. \[ \lim_{x \to \frac{1}{3}^+}\left(\frac{3 x - 1}{\sin{\left(3 x - 1 \right)}}\right) \]
   limitStart with the limit of the given function.✓ Proved
2. \[ = \lim_{x \to \frac{1}{3}^+}\left(\frac{\frac{d}{d x} \left(3 x - 1\right)}{\frac{d}{d x} \sin{\left(3 x - 1 \right)}}\right) \]
   lhopitalThe limit is an indeterminate form 0/0, so apply L'Hopital's rule.✓ Proved
3. \[ = \lim_{x \to \frac{1}{3}^+} \frac{1}{\cos{\left(3 x - 1 \right)}} \]
   simplify simplifyCompute the derivatives of the numerator and denominator. Simplify the fraction by canceling the common factor 3.✓ Proved
4. \[ = \lim_{x \to \frac{1}{3}^+} 1 \]
   substitutionEvaluate the expression inside the limit at x = 1/3.✓ Proved
5. \[ = 1 \]
   limitThe final value of the limit is 1.✓ Proved