Limit of \( \displaystyle \frac{\sin^{2}{\left(x \right)}}{x^{2}} \) as \( x \to 0 \)

Problem 1.59 · easy

1. \[ \lim_{x \to 0^+}\left(\frac{\sin^{2}{\left(x \right)}}{x^{2}}\right) \]
   limit algebraStart with the original limit expression. Rewrite the squared fraction as the square of the fraction.✓ Proved
2. \[ = \left(\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{x}\right)\right)^{2} \]
   limit-lawApply the power rule for limits.✓ Proved
3. \[ = \left(\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(x \right)}}{\frac{d}{d x} x}\right)\right)^{2} \]
   lhopitalUse L'Hôpital's Rule because the limit is an indeterminate form 0/0.✓ Proved
4. \[ = \left(\lim_{x \to 0^+} \cos{\left(x \right)}\right)^{2} \]
   simplifyEvaluate the derivatives.✓ Proved
5. \[ = 1 \]
   limit simplifyEvaluate the limit by substituting x = 0. Calculate the final value.✓ Proved