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

Problem 1.244 · medium

1. \[ \lim_{x \to \infty}\left(\frac{\sqrt{\left(3 x - 1\right)^{2} + 1}}{3 x - 1}\right) \]
   limitEvaluate the limit as x approaches infinity.✓ Proved
2. \[ = \lim_{x \to \infty}\left(\frac{\sqrt{1 + \frac{1}{\left(3 x - 1\right)^{2}}} \left|{3 x - 1}\right|}{3 x - 1}\right) \]
   algebra algebra simplifyFactor out (3*x - 1)**2 from the radicand. Use the property sqrt(a*b) = sqrt(a)*sqrt(b). Simplify the square root of the squared term.✓ Proved
3. \[ = \lim_{x \to \infty} \sqrt{1 + \frac{1}{\left(3 x - 1\right)^{2}}} \]
   simplifySince x approaches infinity, 3*x - 1 is positive, so abs(3*x - 1)/(3*x - 1) = 1.✓ Proved
4. \[ = \lim_{x \to \infty} 1 \]
   limit-lawThe term 1/(3*x - 1)**2 approaches 0 as x approaches infinity.✓ Proved
5. \[ = 1 \]
   limitEvaluate the final expression.✓ Proved