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

Problem 1.157 · medium

1. \[ \lim_{x \to \infty}\left(\frac{2 x + 2}{\sqrt{\left(2 x + 1\right)^{2} + 4}}\right) \]
   limitStart with the original limit expression.✓ Proved
2. \[ = \lim_{x \to \infty}\left(\frac{2 x + 2}{\sqrt{4 x^{2} + 4 x + 5}}\right) \]
   algebra simplifyExpand the square inside the square root. Combine the constant terms inside the square root.✓ Proved
3. \[ = \lim_{x \to \infty}\left(\frac{2 + \frac{2}{x}}{\sqrt{\frac{4 x^{2} + 4 x + 5}{x^{2}}}}\right) \]
   rewriteDivide numerator and denominator by x (noting that for x > 0, x = sqrt(x**2)).✓ Proved
4. \[ = \lim_{x \to \infty}\left(\frac{2 + \frac{2}{x}}{\sqrt{4 + \frac{4}{x} + \frac{5}{x^{2}}}}\right) \]
   algebraDistribute the division by x into the terms.✓ Proved
5. \[ = \lim_{x \to \infty}\left(2 + \frac{2}{x}\right) \left(\lim_{x \to \infty} \sqrt{4 + \frac{4}{x} + \frac{5}{x^{2}}}\right)^{-1} \]
   limit-lawApply the quotient rule for limits.✓ Proved
6. \[ = 1 \]
   limit simplify simplifyEvaluate the limit of each part as x approaches infinity, noting that terms with 1/x vanish. Simplify the square root. Final evaluation.✓ Proved