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

Problem 1.98 · medium

1. \[ \lim_{x \to \infty}\left(\frac{x + 1}{\sqrt{x^{2} + 4}}\right) \]
   limitStart with the given limit.✓ Proved
2. \[ = \lim_{x \to \infty}\left(\frac{1 + \frac{1}{x}}{\sqrt{1 + \frac{4}{x^{2}}}}\right) \]
   algebra simplifyDivide the numerator and denominator by x. Simplify the expression inside the limit.✓ Proved
3. \[ = \lim_{x \to \infty}\left(1 + \frac{1}{x}\right) \left(\lim_{x \to \infty} \sqrt{1 + \frac{4}{x^{2}}}\right)^{-1} \]
   limit-lawApply the quotient rule for limits.✓ Proved
4. \[ = 1 \]
   limit simplifyEvaluate the limits of the numerator and denominator separately. Final result.✓ Proved