\lim_{(x + 5) \to 0} \: ( \frac{1 – \cos(x + 5) }{ {x}^{2} + 10x + 25 } )
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limit 0/0
lim x→0 sin ax/bx = a/b
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lim (x+5)→0 (1 – cos (x + 5)) / (x² + 10x + 25)
= lim (x+5)→0 (1 – cos (x + 5)) / (x + 5)²
= lim a→0 (1 – cos a)/a²
= lim a→0 (2 sin² (a/2))/a²
= 2 ((1/2) / 1)²
= 2 × 1/4
= 1/2