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

Problem 1.42 · hard

1. \[ \lim_{x \to \infty}\left(\frac{3 \left(2 x - 1\right)^{2} + 1}{\left(2 x - 1\right)^{2} - 4}\right) \]
   limitEvaluate the limit of the given function as x approaches infinity.✓ Proved
2. \[ = \lim_{x \to \infty}\left(\frac{12 x^{2} - 12 x + 4}{4 x^{2} - 4 x - 3}\right) \]
   algebra simplifyExpand the squared binomials. Distribute the 3 and combine constant terms.✓ Proved
3. \[ = \lim_{x \to \infty}\left(\frac{12 x^{2}}{4 x^{2} - 4 x - 3} - \frac{12 x}{4 x^{2} - 4 x - 3} + \frac{4}{4 x^{2} - 4 x - 3}\right) \]
   limit-lawSplit the fraction into three separate parts.✓ Proved
4. \[ = - \lim_{x \to \infty}\left(\frac{12 x}{4 x^{2} - 4 x - 3}\right) + \lim_{x \to \infty}\left(\frac{12 x^{2}}{4 x^{2} - 4 x - 3}\right) + \lim_{x \to \infty}\left(\frac{4}{4 x^{2} - 4 x - 3}\right) \]
   limit-lawApply the limit law for sums and differences.✓ Proved
5. \[ = \lim_{x \to \infty} 3 - \lim_{x \to \infty}\left(\frac{12 x}{4 x^{2} - 4 x - 3}\right) + \lim_{x \to \infty}\left(\frac{4}{4 x^{2} - 4 x - 3}\right) \]
   simplifyDivide the numerator and denominator of the first term by x**2.✓ Proved
6. \[ = 3 - \lim_{x \to \infty}\left(\frac{12 x}{4 x^{2} - 4 x - 3}\right) + \lim_{x \to \infty}\left(\frac{4}{4 x^{2} - 4 x - 3}\right) \]
   simplifySimplify the first limit term.✓ Proved
7. \[ = 3 \]
   limit simplifyEvaluate the remaining limits as x approaches infinity. Final result.✓ Proved