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

Problem 1.167 · hard

1. \[ \lim_{x \to \infty}\left(\frac{4 \left(2 x + 1\right)^{2} - 1}{2 x + 2 \left(2 x + 1\right)^{2} + 1}\right) \]
   limit algebraStart with the limit of the given function. Rearrange the denominator terms.✓ Proved
2. \[ = \lim_{x \to \infty}\left(\frac{4 \left(2 x + 1\right)^{2} - 1}{8 x^{2} + 10 x + 3}\right) \]
   simplify simplify simplifyExpand the squared term in the denominator. Distribute the 2 in the denominator. Combine like terms in the denominator.✓ Proved
3. \[ = \lim_{x \to \infty}\left(\frac{16 x^{2} + 16 x + 3}{8 x^{2} + 10 x + 3}\right) \]
   simplify simplifyExpand the squared term in the numerator. Simplify the numerator.✓ Proved
4. \[ = \lim_{x \to \infty}\left(\frac{16 x^{2}}{8 x^{2} + 10 x + 3} + \frac{16 x}{8 x^{2} + 10 x + 3} + \frac{3}{8 x^{2} + 10 x + 3}\right) \]
   limit-lawSplit the fraction into three parts.✓ Proved
5. \[ = \lim_{x \to \infty}\left(\frac{16 x}{8 x^{2} + 10 x + 3}\right) + \lim_{x \to \infty}\left(\frac{16 x^{2}}{8 x^{2} + 10 x + 3}\right) + \lim_{x \to \infty}\left(\frac{3}{8 x^{2} + 10 x + 3}\right) \]
   limit-lawApply the sum rule for limits.✓ Proved
6. \[ = 2 \]
   limitEvaluate each limit term separately.✓ Proved