Limit of \( \displaystyle \frac{- 3 x + e^{3 x} - 1}{9 x^{2}} \) as \( x \to 0 \)

Problem 1.74 · medium

1. \[ \lim_{x \to 0^+}\left(\frac{- 3 x + e^{3 x} - 1}{9 x^{2}}\right) \]
   limit algebraIdentify the limit to be evaluated. Rearrange the terms in the numerator.✓ Proved
2. \[ = \lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- 3 x + e^{3 x} - 1\right)}{\frac{d}{d x} 9 x^{2}}\right) \]
   lhopitalApply L'Hépital's rule because the limit is an indeterminate form 0/0.✓ Proved
3. \[ = \lim_{x \to 0^+}\left(\frac{3 e^{3 x} - 3}{18 x}\right) \]
   simplify factorCompute the derivatives. Factor out the constant 3 from the numerator.✓ Proved
4. \[ = \lim_{x \to 0^+}\left(\frac{e^{3 x} - 1}{6 x}\right) \]
   simplifySimplify the fraction by dividing 3/18.✓ Proved
5. \[ = \lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(e^{3 x} - 1\right)}{\frac{d}{d x} 6 x}\right) \]
   lhopitalApply L'Hépital's rule again as it is still 0/0.✓ Proved
6. \[ = \lim_{x \to 0^+}\left(\frac{e^{3 x}}{2}\right) \]
   simplify simplifyCompute the derivatives. Simplify the fraction 3/6 to 1/2.✓ Proved
7. \[ = \frac{1}{2} \]
   limit simplifyEvaluate the limit by substituting x = 0. Final result.✓ Proved