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

Problem 1.94 · 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 original function.✓ Proved
2. \[ = \lim_{x \to \frac{1}{3}^+}\left(\frac{\frac{d}{d x} \left(e^{3 x - 1} - 1\right)}{\frac{d}{d x} \left(3 x - 1\right)}\right) \]
   lhopitalApply L'Hopital's rule because the limit is an indeterminate form 0/0.✓ Proved
3. \[ = \lim_{x \to \frac{1}{3}^+} e^{3 x - 1} \]
   simplify simplifyDifferentiate the numerator and the denominator. Simplify the fraction by canceling the constant 3.✓ Proved
4. \[ = 1 \]
   substitution simplify simplifySubstitute the value x = 1/3 into the expression. Simplify the exponent. Evaluate the final exponential expression.✓ Proved