Limit of \( \displaystyle \frac{4 x - 1}{\left(2 x - 1\right)^{2} + 3} \) as \( x \to \infty \)

Problem 1.418 · medium

1. \[ \lim_{x \to \infty}\left(\frac{4 x - 1}{\left(2 x - 1\right)^{2} + 3}\right) \]
   limitStart with the original limit.✓ Proved
2. \[ = \lim_{x \to \infty}\left(\frac{4 x - 1}{4 x^{2} - 4 x + 4}\right) \]
   algebra simplifyExpand the denominator. Combine the constant terms.✓ Proved
3. \[ = \lim_{x \to \infty}\left(\frac{\frac{4}{x} - \frac{1}{x^{2}}}{4 - \frac{4}{x} + \frac{4}{x^{2}}}\right) \]
   rewriteDivide numerator and denominator by x**2.✓ Proved
4. \[ = \lim_{x \to \infty}\left(\frac{4}{x} - \frac{1}{x^{2}}\right) \left(\lim_{x \to \infty}\left(4 - \frac{4}{x} + \frac{4}{x^{2}}\right)\right)^{-1} \]
   limit-lawApply the quotient rule for limits.✓ Proved
5. \[ = 0 \]
   limit simplifyEvaluate the limits of the numerator and denominator. Final result.✓ Proved