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

Problem 1.99 · 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. \[ = \lim_{x \to - \frac{1}{3}^+} \text{NaN} \]
   limitIdentify the indeterminate form 0/0.Not checked
3. \[ = \lim_{x \to - \frac{1}{3}^+}\left(\frac{\frac{d}{d x} \left(3 x + 1\right)}{\frac{d}{d x} \sin{\left(3 x + 1 \right)}}\right) \]
   lhopitalApply L'Hopital's Rule because the limit is of the form 0/0.Not checked
4. \[ = \lim_{x \to - \frac{1}{3}^+} \frac{1}{\cos{\left(3 x + 1 \right)}} \]
   simplify simplifyCompute the derivatives of the numerator and denominator. Cancel the common factor of 3.✓ Proved
5. \[ = \lim_{x \to - \frac{1}{3}^+} 1 \]
   substitution simplifyEvaluate the expression at the point x = -1/3. Evaluate the cosine term.✓ Proved
6. \[ = 1 \]
   simplifyFinal result.✓ Proved