Limit of \( \displaystyle \frac{e^{4 x - 2} - 1}{\sin{\left(2 x - 1 \right)}} \) as \( x \to \frac{1}{2} \)

Problem 1.313 · medium

1. \[ \lim_{x \to \frac{1}{2}^+}\left(\frac{e^{4 x - 2} - 1}{\sin{\left(2 x - 1 \right)}}\right) \]
   limitStart with the original limit.✓ Proved
2. \[ = \lim_{x \to \frac{1}{2}^+}\left(\frac{\frac{d}{d x} \left(e^{4 x - 2} - 1\right)}{\frac{d}{d x} \sin{\left(2 x - 1 \right)}}\right) \]
   lhopitalApply L'Hopital's rule because the limit is of the form 0/0.✓ Proved
3. \[ = \lim_{x \to \frac{1}{2}^+}\left(\frac{2 e^{4 x - 2}}{\cos{\left(2 x - 1 \right)}}\right) \]
   simplify simplifyCompute the derivatives. Simplify the fraction by dividing by 2.✓ Proved
4. \[ = \lim_{x \to \frac{1}{2}^+} 2 \]
   limit simplify simplifyEvaluate the limit by substituting x = 1/2. Simplify the exponents and trigonometric arguments. Evaluate the exponential and cosine functions.✓ Proved
5. \[ = 2 \]
   simplifyFinal result.✓ Proved