Limit of \( \displaystyle \frac{e^{4 x} - 1}{\sin{\left(2 x \right)}} \) as \( x \to 0 \)

Problem 1.232 · medium

1. \[ \lim_{x \to 0^+}\left(\frac{e^{4 x} - 1}{\sin{\left(2 x \right)}}\right) \]
   limitStart with the limit of the given function.✓ Proved
2. \[ = \lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(e^{4 x} - 1\right)}{\frac{d}{d x} \sin{\left(2 x \right)}}\right) \]
   lhopitalApply L'Hôpital's rule because the limit is an indeterminate form 0/0.✓ Proved
3. \[ = \lim_{x \to 0^+}\left(\frac{2 e^{4 x}}{\cos{\left(2 x \right)}}\right) \]
   simplify simplify limitCompute the derivatives in the numerator and denominator. Simplify the fraction by canceling the common factor of 2. The expression is now continuous at x = 0.✓ Proved
4. \[ = 2 \]
   substitution simplify simplifySubstitute x = 0 into the expression. Evaluate the exponential and trigonometric functions at 0. Final result.✓ Proved