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

Problem 1.308 · medium

1. \[ \lim_{x \to - \frac{1}{2}^+}\left(\frac{e^{2 x + 1} - 1}{2 x + 1}\right) \]
   limitSet up the limit with the specified direction.✓ 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 0/0.✓ Proved
3. \[ = \lim_{x \to - \frac{1}{2}^+} e^{2 x + 1} \]
   simplify simplifyCompute the derivatives. Simplify the fraction.✓ Proved
4. \[ = 1 \]
   limit algebra simplifyEvaluate the limit by substituting the value of x. Simplify the exponent. Final result.✓ Proved