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

Problem 1.170 · medium

1. \[ \lim_{x \to 0^+}\left(\frac{\cos{\left(2 x \right)} - 1}{e^{2 x} - 1}\right) \]
   limitStart with the limit of the original function.✓ Proved
2. \[ = \lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(\cos{\left(2 x \right)} - 1\right)}{\frac{d}{d x} \left(e^{2 x} - 1\right)}\right) \]
   lhopitalApply L'Hopital's rule since the limit is an indeterminate form 0/0.✓ Proved
3. \[ = \lim_{x \to 0^+}\left(- e^{- 2 x} \sin{\left(2 x \right)}\right) \]
   simplify simplifyCompute the derivatives. Cancel the common factor of 2.✓ Proved
4. \[ = \lim_{x \to 0^+}\left(- \sin{\left(2 x \right)}\right) \left(\lim_{x \to 0^+} e^{2 x}\right)^{-1} \]
   limit-lawApply the quotient rule for limits.✓ Proved
5. \[ = 0 \]
   limit simplifyEvaluate the individual limits. Final result.✓ Proved