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

Problem 1.100 · medium

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