Nice-limit-For-1-lt-a-lt-1-the-value-of-lim-x-a-x-2-2ax-a-2-x-2-ax-a-2-1-x-2-1-a-2-2-

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

Nice limit!

For - 1 < a < 1 the value of \lim_{x \to - a} \frac{(x^{2} + 2 a x + a^{2}) (x^{2} + a x + a^{2})}{{(\sqrt{1 - x^{2}} - \sqrt{1 - a^{2}})}^{2}} = ?

Answered by liberty last updated on 10/Jan/21

\lim_{x \to - a} \frac{{(x + a)}^{2} (a^{2} - a^{2} + a^{2})}{{(\sqrt{1 - x^{2}} - \sqrt{1 - a^{2}})}^{2}} = a^{2} \lim_{x \to - a} {(\frac{(x + a) (\sqrt{1 - x^{2}} + \sqrt{1 - a^{2}})}{a^{2} - x^{2}})}^{2}

= a^{2} \lim_{x \to - a} {(\frac{\sqrt{1 - x^{2}} + \sqrt{1 - a^{2}}}{a - x})}^{2}

= a^{2} \times {(\frac{2 \sqrt{1 - a^{2}}}{2 a})}^{2} = 1 - a^{2}

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