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Question Number 128821 by benjo_mathlover last updated on 10/Jan/21

Nice limit ! For −1<a <1 the value of lim_(x→−a) (((x^2 +2ax+a^2 )(x^2 +ax+a^2 ))/(((√(1−x^2 ))−(√(1−a^2 )) )^2 )) =?

Nicelimit! For−1<a<1thevalueoflimx→−a(x2+2ax+a2)(x2+ax+a2)(1−x2−1−a2)2=?

Answered by liberty last updated on 10/Jan/21

lim_(x→−a) (((x+a)^2 (a^2 −a^2 +a^2 ))/(((√(1−x^2 ))−(√(1−a^2 )) )^2 )) = a^2 lim_(x→−a) ((((x+a)((√(1−x^2 ))+(√(1−a^2 ))))/(a^2 −x^2 )))^2 = a^2 lim_(x→−a) ((((√(1−x^2 )) +(√(1−a^2 )))/(a−x)))^2 = a^2 × (((2(√(1−a^2 )))/(2a)) )^2 = 1−a^2

limx→−a(x+a)2(a2−a2+a2)(1−x2−1−a2)2=a2limx→−a((x+a)(1−x2+1−a2)a2−x2)2 =a2limx→−a(1−x2+1−a2a−x)2 =a2×(21−a22a)2=1−a2

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