Limit of \( \displaystyle \frac{- 2 x + e^{2 x} - 1}{4 x^{2}} \) as \( x \to 0 \)

Problem 1.81 · medium

1. \[ \lim_{x \to 0^+}\left(\frac{- 2 x + e^{2 x} - 1}{4 x^{2}}\right) \]
   limit algebraStarting with the original limit expression. Rearranging the terms in the numerator.✓ Proved
2. \[ = \lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- 2 x + e^{2 x} - 1\right)}{\frac{d}{d x} 4 x^{2}}\right) \]
   lhopitalApplying L'Hopital's rule because the limit is of the form 0/0.✓ Proved
3. \[ = \lim_{x \to 0^+}\left(\frac{2 e^{2 x} - 2}{8 x}\right) \]
   simplify factorComputing the derivatives in the numerator and denominator. Factoring out the constant 2 from the numerator.✓ Proved
4. \[ = \lim_{x \to 0^+}\left(\frac{e^{2 x} - 1}{4 x}\right) \]
   simplifySimplifying the fraction by dividing the constants.✓ Proved
5. \[ = \lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(e^{2 x} - 1\right)}{\frac{d}{d x} 4 x}\right) \]
   lhopitalApplying L'Hopital's rule again as the limit is still 0/0.✓ Proved
6. \[ = \lim_{x \to 0^+}\left(\frac{e^{2 x}}{2}\right) \]
   simplify simplifyComputing the derivatives. Simplifying the expression.✓ Proved
7. \[ = \frac{1}{2} \]
   limitEvaluating the limit at x = 0.✓ Proved
8. \[ = 0.5 \]
   simplifyFinal result.✓ Proved