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

Problem 1.306 · hard

1. \[ \lim_{x \to \infty}\left(\frac{4 \left(x - 1\right)^{2} - 1}{x + 2 \left(x - 1\right)^{2} - 1}\right) \]
   limitEvaluate the limit of the given function at infinity.✓ Proved
2. \[ = \lim_{x \to \infty}\left(\frac{4 x^{2} - 8 x + 3}{2 x^{2} - 3 x + 1}\right) \]
   algebra simplifyExpand the squared terms. Simplify the numerator and denominator.✓ Proved
3. \[ = \lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(4 x^{2} - 8 x + 3\right)}{\frac{d}{d x} \left(2 x^{2} - 3 x + 1\right)}\right) \]
   lhopitalApply L'Hopital's rule because the limit is of the form infinity/infinity.✓ Proved
4. \[ = \lim_{x \to \infty}\left(\frac{8 x - 8}{4 x - 3}\right) \]
   simplifyCompute the derivatives.✓ Proved
5. \[ = \lim_{x \to \infty}\left(\frac{\frac{8}{x} - \frac{8}{x^{2}}}{\frac{4}{x} - \frac{3}{x^{2}}}\right) \]
   algebraDivide both numerator and denominator by x.✓ Proved
6. \[ = \lim_{x \to \infty}\left(\frac{8}{x}\right) \left(\lim_{x \to \infty}\left(\frac{4}{x}\right)\right)^{-1} \]
   limit-lawApply the quotient rule for limits, noting that the individual limits are 0.Not checked
7. \[ = \lim_{x \to \infty} 8 \left(\lim_{x \to \infty} 4\right)^{-1} \]
   limit-lawMultiply numerator and denominator by x to resolve the 0/0 form.Not checked
8. \[ = 2 \]
   limitEvaluate the resulting limit.✓ Proved