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

Problem 1.356 · medium

1. \[ \lim_{x \to \frac{1}{2}^+}\left(\frac{\tan{\left(2 x - 1 \right)}}{2 x - 1}\right) \]
   limitStart with the original limit.✓ Proved
2. \[ = \lim_{u \to 0^+}\left(\frac{\tan{\left(u \right)}}{u}\right) \]
   substitutionLet u = 2*x - 1. 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'Hopital's Rule since the limit is in the form 0/0.✓ Proved
4. \[ = \lim_{u \to 0^+} \sec^{2}{\left(u \right)} \]
   simplifyCompute the derivatives of the numerator and denominator.✓ Proved
5. \[ = 1 \]
   limit simplifyEvaluate the limit by plugging in u = 0. Simplify the final value.✓ Proved