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

Problem 1.256 · medium

1. \[ \lim_{x \to \frac{1}{2}^+}\left(\frac{e^{2 x - 1} - 1}{2 x - 1}\right) \]
   limitSet up the limit of the function as x approaches 1/2.✓ Proved
2. \[ = \lim_{x \to \frac{1}{2}^+}\left(\frac{\frac{d}{d x} \left(e^{2 x - 1} - 1\right)}{\frac{d}{d x} \left(2 x - 1\right)}\right) \]
   lhopitalApply L'Hôpital's rule since the limit is an indeterminate form of type 0/0.✓ Proved
3. \[ = \lim_{x \to \frac{1}{2}^+} e^{2 x - 1} \]
   simplify simplifyCompute the derivatives of the numerator and denominator. Simplify the fraction by canceling the constant 2.✓ Proved
4. \[ = 1 \]
   substitution algebra simplifySubstitute x = 1/2 into the expression. Simplify the exponent. Evaluate the exponential function.✓ Proved