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

Problem 1.124 · medium

1. \[ \lim_{x \to \infty}\left(\frac{x}{\sqrt{\left(x - 1\right)^{2} + 4}}\right) \]
   limitStart with the given limit.✓ Proved
2. \[ = \lim_{x \to \infty}\left(\frac{x}{\sqrt{x^{2} - 2 x + 5}}\right) \]
   algebra simplifyExpand the squared term inside the square root. Combine the constant terms.✓ Proved
3. \[ = \lim_{x \to \infty}\left(\frac{x}{\sqrt{1 - \frac{2}{x} + \frac{5}{x^{2}}} \left|{x}\right|}\right) \]
   rewriteFactor out x**2 from the expression inside the square root.✓ Proved
4. \[ = \lim_{x \to \infty} \frac{1}{\sqrt{1 - \frac{2}{x} + \frac{5}{x^{2}}}} \]
   algebra simplifyTake x out of the square root, noting that x is positive as x approaches infinity. Cancel the common factor of x in the numerator and denominator.✓ Proved
5. \[ = \lim_{x \to \infty} 1 \]
   limit-lawApply the limit to each term inside the expression.✓ Proved
6. \[ = 1 \]
   limitEvaluate the final expression.✓ Proved