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

Problem 1.430 · medium

1. \[ \lim_{x \to - \frac{1}{3}^+}\left(\frac{e^{3 x + 1} - 1}{3 x + 1}\right) \]
   limitStart with the limit of the function as x approaches -1/3 from the right.✓ Proved
2. \[ = \lim_{x \to - \frac{1}{3}^+}\left(\frac{\frac{d}{d x} \left(e^{3 x + 1} - 1\right)}{\frac{d}{d x} \left(3 x + 1\right)}\right) \]
   lhopitalThe limit is an indeterminate form 0/0, so apply l'Hopital's rule.✓ Proved
3. \[ = \lim_{x \to - \frac{1}{3}^+} e^{3 x + 1} \]
   simplify simplifyCalculate the derivatives of the numerator and denominator. Simplify the fraction by canceling the constant 3.✓ Proved
4. \[ = 1 \]
   limit simplify simplifyEvaluate the limit by substituting x = -1/3. Simplify the exponent. Final evaluation.✓ Proved