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

Problem 1.330 · medium

1. \[ \lim_{x \to 1^+}\left(\frac{e^{2 x - 2} - 1}{\sin{\left(x - 1 \right)}}\right) \]
   limit algebraEvaluate the limit of the function as x approaches 1. Rewrite the exponent to reveal the common term (x - 1).✓ Proved
2. \[ = \lim_{x \to 1^+}\left(\frac{\frac{d}{d x} \left(e^{2 x - 2} - 1\right)}{\frac{d}{d x} \sin{\left(x - 1 \right)}}\right) \]
   lhopitalApply L'Hopital's rule since the limit results in an indeterminate form 0/0.✓ Proved
3. \[ = \lim_{x \to 1^+}\left(\frac{2 e^{2 x - 2}}{\cos{\left(x - 1 \right)}}\right) \]
   simplifyCompute the derivatives of the numerator and denominator.✓ Proved
4. \[ = 2 \]
   limit algebra simplify simplifyEvaluate the limit by substituting x = 1. Simplify the arguments of the exponential and trigonometric functions. Evaluate the exponential and trigonometric values. Final result.✓ Proved