Limit of \( \displaystyle \frac{e^{2 x} - 1}{2 x} \) as \( x \to 0 \)

Problem 1.139 · medium

1. \[ \lim_{x \to 0^+}\left(\frac{e^{2 x} - 1}{2 x}\right) \]
   limitStart with the limit of the function as x approaches 0 from the right.✓ Proved
2. \[ = \lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(e^{2 x} - 1\right)}{\frac{d}{d x} 2 x}\right) \]
   lhopitalApply L'Hopital's rule because the limit is an indeterminate form of type 0/0.✓ Proved
3. \[ = \lim_{x \to 0^+} e^{2 x} \]
   simplify simplifyCompute the derivatives of the numerator and the denominator. Simplify the expression by canceling the constant 2.✓ Proved
4. \[ = 1 \]
   limitEvaluate the limit as x approaches 0.✓ Proved