Limit of \( \displaystyle \frac{\ln{\left(2 x + 1 \right)}}{2 x + 1} \) as \( x \to \infty \)

Problem 1.88 · medium

1. \[ \lim_{x \to \infty}\left(\frac{\ln{\left(2 x + 1 \right)}}{2 x + 1}\right) \]
   limitStart with the original limit.✓ Proved
2. \[ = \lim_{u \to \infty}\left(\frac{\ln{\left(u \right)}}{u}\right) \]
   substitution limitLet u = 2*x + 1. As x approaches infinity, u also approaches infinity. The limit is now expressed in terms of u.✓ Proved
3. \[ = \lim_{u \to \infty}\left(\frac{\frac{d}{d u} \ln{\left(u \right)}}{\frac{d}{d u} u}\right) \]
   lhopitalApply L'Hopital's rule since the limit is of the form infinity/infinity.✓ Proved
4. \[ = \lim_{u \to \infty} \frac{1}{u} \]
   simplify simplifyCompute the derivatives of the numerator and denominator. Simplify the fraction.✓ Proved
5. \[ = 0 \]
   limitEvaluate the final limit.✓ Proved