Limit of \( \displaystyle \frac{6 x - 1}{\left(3 x - 1\right)^{2} + 3} \) as \( x \to \infty \)

Problem 1.323 · medium

1. \[ \lim_{x \to \infty}\left(\frac{6 x - 1}{\left(3 x - 1\right)^{2} + 3}\right) \]
   limitStart with the limit of the given function as x approaches infinity.✓ Proved
2. \[ = \lim_{x \to \infty}\left(\frac{6 x - 1}{9 x^{2} - 6 x + 4}\right) \]
   simplifyExpand the denominator.✓ Proved
3. \[ = \lim_{x \to \infty}\left(\frac{\frac{6}{x} - \frac{1}{x^{2}}}{9 - \frac{6}{x} + \frac{4}{x^{2}}}\right) \]
   algebra simplifyDivide numerator and denominator by x**2. Simplify the fractions inside the limit.✓ Proved
4. \[ = \lim_{x \to \infty} 0 \]
   limit-law simplifyEvaluate the limit of each term as x approaches infinity. Simplify the resulting fraction.✓ Proved
5. \[ = 0 \]
   limitThe final value of the limit.✓ Proved