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

Problem 1.371 · hard

1. \[ \lim_{x \to 1^+}\left(\frac{- 2 x + \left(2 x - 1\right)^{2} + 1}{\left(2 x - 1\right)^{2} - 1}\right) \]
   limitEvaluate the limit of the given function at x = 1.✓ Proved
2. \[ = \lim_{x \to 1^+}\left(\frac{\frac{d}{d x} \left(- 2 x + \left(2 x - 1\right)^{2} + 1\right)}{\frac{d}{d x} \left(\left(2 x - 1\right)^{2} - 1\right)}\right) \]
   lhopitalApply L'Hopital's Rule because the limit is in the form 0/0.✓ Proved
3. \[ = \lim_{x \to 1^+}\left(\frac{8 x - 6}{8 x - 4}\right) \]
   simplify simplifyCompute the derivatives of the numerator and denominator. Simplify the expression by distributing the constants.✓ Proved
4. \[ = \lim_{x \to 1^+}\left(1 - \frac{2}{8 x - 4}\right) \]
   algebraSplit the fraction into two parts.✓ Proved
5. \[ = \lim_{x \to 1^+}\left(1 - \frac{1}{4 x - 2}\right) \]
   simplifySimplify the fraction coefficient.✓ Proved
6. \[ = \lim_{x \to 1^+} \frac{1}{2} \]
   substitution simplifySubstitute x = 1 into the expression. Evaluate the denominator.✓ Proved
7. \[ = \frac{1}{2} \]
   limitFinal result.✓ Proved