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

Problem 1.147 · medium

1. \[ \lim_{x \to \infty}\left(\frac{\sqrt{4 x^{2} + 1}}{2 x}\right) \]
   limitIdentify the limit to evaluate.✓ Proved
2. \[ = \lim_{x \to \infty}\left(\frac{\sqrt{4 + \frac{1}{x^{2}}} \left|{x}\right|}{2 x}\right) \]
   factor algebra simplifyFactor out x**2 from the radicand. Apply the property sqrt(a*b) = sqrt(a)*sqrt(b). Simplify the square root of x**2.✓ Proved
3. \[ = \lim_{x \to \infty}\left(\frac{\sqrt{4 + \frac{1}{x^{2}}}}{2}\right) \]
   simplify cancelSince x approaches infinity, |x| = x. Cancel the common factor of x in the numerator and denominator.✓ Proved
4. \[ = \frac{\lim_{x \to \infty} 2}{2} \]
   limit-lawApply the limit law for a quotient.✓ Proved
5. \[ = 1 \]
   simplify simplifyEvaluate the limit of the constant term. Final simplification.✓ Proved