Limit of \( \displaystyle \frac{\sqrt{x^{2} + 9} - 3}{x^{2}} \) as \( x \to 0 \)

Problem 1.52 · medium

1. \[ \lim_{x \to 0^+}\left(\frac{\sqrt{x^{2} + 9} - 3}{x^{2}}\right) \]
   limit rationalizeStart with the initial limit expression. Multiply the numerator and denominator by the conjugate of the numerator.✓ Proved
2. \[ = \lim_{x \to 0^+} \frac{1}{\sqrt{x^{2} + 9} + 3} \]
   algebra simplify cancelExpand the product in the numerator. Simplify the numerator. Cancel the common x**2 term in the numerator and denominator.✓ Proved
3. \[ = \frac{1}{6} \]
   limit simplify simplifyEvaluate the limit by substituting x = 0. Simplify the square root. Final simplification.✓ Proved