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

Problem 1.156 · medium

1. \[ \lim_{x \to - \frac{1}{2}^+}\left(\frac{\tan{\left(2 x + 1 \right)}}{2 x + 1}\right) \]
   limit algebraEvaluate the limit of the function as x approaches -1/2. Rewrite the expression to isolate the term approaching zero.✓ Proved
2. \[ = \lim_{u \to 0^+}\left(\frac{\tan{\left(u \right)}}{u}\right) \]
   substitutionLet u = 2*(x + 1/2), so as x approaches -1/2, u approaches 0.✓ Proved
3. \[ = \lim_{u \to 0^+}\left(\frac{\frac{d}{d u} \tan{\left(u \right)}}{\frac{d}{d u} u}\right) \]
   lhopitalApply L'Hôpital's rule since the limit is an indeterminate form of 0/0.✓ Proved
4. \[ = \lim_{u \to 0^+} \sec^{2}{\left(u \right)} \]
   simplifyCompute the derivatives of the numerator and denominator.✓ Proved
5. \[ = 1 \]
   limitEvaluate the limit as u approaches 0.✓ Proved