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

Problem 1.379 · hard

1. \[ \lim_{x \to \infty}\left(\frac{4 \left(3 x + 1\right)^{2} - 1}{3 x + 2 \left(3 x + 1\right)^{2} + 1}\right) \]
   limitStart with the limit of the given function.✓ Proved
2. \[ = \lim_{x \to \infty}\left(\frac{4 \left(3 x + 1\right)^{2} - 1}{3 x + 2 \left(3 x + 1\right)^{2} + 2}\right) \]
   algebraRearrange the denominator terms.✓ Proved
3. \[ = \lim_{x \to \infty}\left(\frac{36 x^{2} + 24 x + 3}{18 x^{2} + 15 x + 4}\right) \]
   simplify algebra simplifyExpand the squared binomial term. Distribute the constants into the parentheses. Combine like terms in the numerator and denominator.✓ Proved
4. \[ = \lim_{x \to \infty}\left(\frac{36 + \frac{24}{x} + \frac{3}{x^{2}}}{18 + \frac{15}{x} + \frac{4}{x^{2}}}\right) \]
   algebra simplifyDivide the numerator and denominator by x**2. Simplify the fractions within the expression.✓ Proved
5. \[ = \lim_{x \to \infty} 2 \]
   limit-lawApply the limit laws for the quotient of limits.✓ Proved
6. \[ = 2 \]
   limitEvaluate the final numerical value.✓ Proved