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

Problem 1.465 · medium

1. \[ \lim_{x \to \infty}\left(\frac{3 x}{\sqrt{\left(3 x - 1\right)^{2} + 4}}\right) \]
   limitStart with the original limit expression.✓ Proved
2. \[ = \lim_{x \to \infty}\left(\frac{3 x}{\sqrt{9 x^{2} - 6 x + 5}}\right) \]
   algebra simplifyExpand the squared term inside the square root. Combine the constant terms inside the square root.✓ Proved
3. \[ = \lim_{x \to \infty}\left(\frac{3}{\sqrt{\frac{9 x^{2} - 6 x + 5}{x^{2}}}}\right) \]
   rewrite limit-lawDivide the numerator and denominator by x (noting x > 0). Apply the limit law for the quotient of two functions.✓ Proved
4. \[ = \lim_{x \to \infty}\left(\frac{3}{\sqrt{9 - \frac{6}{x} + \frac{5}{x^{2}}}}\right) \]
   simplifyDistribute the division by x^2 inside the square root.✓ Proved
5. \[ = \lim_{x \to \infty} 1 \]
   infinity simplifyEvaluate the terms involving 1/x as x approaches infinity. Simplify the expression inside the limit.✓ Proved
6. \[ = 1 \]
   limitEvaluate the final numerical value.✓ Proved