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

Problem 1.123 · medium

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