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

Problem 1.138 · medium

1. \[ \lim_{x \to \infty}\left(\frac{3 \left(x - 1\right)^{2} + 1}{\left(x - 1\right)^{2} - 4}\right) \]
   limitStart with the limit of the given function.✓ Proved
2. \[ = \lim_{x \to \infty}\left(\frac{3 x^{2} - 6 x + 4}{x^{2} - 2 x - 3}\right) \]
   algebra simplify simplifyExpand the squared terms. Distribute the 3. Combine constant terms in the numerator and denominator.✓ Proved
3. \[ = \lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(3 x^{2} - 6 x + 4\right)}{\frac{d}{d x} \left(x^{2} - 2 x - 3\right)}\right) \]
   lhopitalApply L'Hopital's rule since the limit is of the form infinity/infinity.✓ Proved
4. \[ = \lim_{x \to \infty}\left(\frac{6 x - 6}{2 x - 2}\right) \]
   simplify factorCompute the derivatives. Factor out common terms.✓ Proved
5. \[ = \lim_{x \to \infty} 3 \]
   simplifyCancel the common factor (x-1).✓ Proved
6. \[ = 3 \]
   limitEvaluate the final limit.✓ Proved