silemus.
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  1. 4L'Hopital's rule, limit
  2. 6Limit of (1−cos⁡x)/x2(1 - \cos x)/x^2
  3. 16Implicit differentiation, folium
  4. 21Maximizing rectangle area
  5. 22Optimizing box volume
  6. 23Related rates, sphere volume
  7. 24Related rates, circle area
  8. 27Integral of ln⁡x\ln x by parts
  9. 28Integration by parts
  10. 30Integral of sin⁡2x\sin^2 x
  11. 35Area between x2x^2 and xx
  12. 37Volume by disk method
  13. 41Telescoping series sum
  14. 44Maclaurin series of cos⁡x\cos x
  15. 45Radius of convergence, ratio test
  16. 49Double integral, inner yy first
  17. 50Separable ODE dy/dx=2ydy/dx = 2y

6Limit of (1−cos⁡x)/x2(1 - \cos x)/x^2

Find lim⁡x→01−cos⁡xx2\displaystyle \lim_{x\to 0} \frac{1 - \cos x}{x^2}.

Use 1 − cos x = 2sin²(x/2) to remove the 0/0 form. Rewriting x² as 4(x/2)² turns the limit into (1/2)·(sin(x/2)/(x/2))², and since sinθ/θ → 1, the limit equals 1/2. Checking x = 0.1 gives ≈0.4996, close to 1/2.

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