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

Problem 1.260 · medium

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