Limit of \( \displaystyle \frac{e^{6 x - 2} - 1}{\sin{\left(3 x - 1 \right)}} \) as \( x \to \frac{1}{3} \)

Problem 1.427 · medium

1. \[ \lim_{x \to \frac{1}{3}^+}\left(\frac{e^{6 x - 2} - 1}{\sin{\left(3 x - 1 \right)}}\right) \]
   limitStart with the original limit.✓ Proved
2. \[ = \lim_{x \to \frac{1}{3}^+}\left(\frac{\frac{d}{d x} \left(e^{6 x - 2} - 1\right)}{\frac{d}{d x} \sin{\left(3 x - 1 \right)}}\right) \]
   lhopitalThe limit is an indeterminate form 0/0, so we apply L'Hôpital's Rule.✓ Proved
3. \[ = \lim_{x \to \frac{1}{3}^+}\left(\frac{2 e^{6 x - 2}}{\cos{\left(3 x - 1 \right)}}\right) \]
   simplify simplifyCompute the derivatives of the numerator and denominator. Simplify the fraction by dividing the constants.✓ Proved
4. \[ = 2 \]
   limit simplify simplify simplifySubstitute the value x = 1/3 into the expression. Simplify the arguments of the exponential and trigonometric functions. Evaluate the exponential and cosine functions. Final result.✓ Proved