lim-x-a-x-1-2-a-1-2-x-a-k-a-gt-0-lim-x-a-x-1-3-a-1-3-x-2-a-2-

Question Number 84283 by jagoll last updated on 11/Mar/20

\lim_{x \to a} \frac{{(∣ x ∣ - 1)}^{2} - {(∣ a ∣ - 1)}^{2}}{x - a} = k, a > 0

\lim_{x \to a} \frac{{(∣ x ∣ - 1)}^{3} - {(∣ a ∣ - 1)}^{3}}{x^{2} - a^{2}} =

Answered by john santu last updated on 11/Mar/20

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

\lim_{x \to a} \frac{x^{2} - 2 x + 1 - a^{2} + 2 a - 1}{x - a} = k

\lim_{x \to a} \frac{(x - a) (x + a) - 2 (x - a)}{x - a} = k

\lim_{x \to a} \frac{(x - a) (x + a - 2)}{x - a} = k

\Rightarrow k = 2 a - 2 \Rightarrow a - 1 = \frac{1}{2} k

(2) \lim_{x \to a} \frac{{(x - 1)}^{3} - {(a - 1)}^{3}}{x^{2} - a^{2}} =

\lim_{x \to a} \frac{3 {(x - 1)}^{2}}{2 x} = \frac{3 {(a - 1)}^{2}}{2 a}

= \frac{3 \times \frac{1}{4} k^{2}}{2 a} = \frac{{3 k}^{2}}{8 a}

Commented by jagoll last updated on 11/Mar/20

thank you mister

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