lim-x-0-x-2-1-x-2-1-2000x-5-2000x-5-

Question Number 128624 by john_santu last updated on 09/Jan/21

\lim_{x \to 0} \frac{∣ x^{2} + 1 ∣ - ∣ x^{2} - 1 ∣}{∣ 2000 x + 5 ∣ - ∣ 2000 x - 5 ∣} = ?

Answered by liberty last updated on 09/Jan/21

\lim_{x \to 0} \frac{(x^{2} + 1) + (x^{2} - 1)}{(2000 x + 5) + (2000 x - 5)} = \lim_{x \to 0} \frac{{2 x}^{2}}{4000 x} = 0

Answered by mathmax by abdo last updated on 09/Jan/21

\lim_{x \to 0_{+}} f (x) = \lim_{x \to 0^{+}} \frac{x^{2} + 1 - (1 - x^{2})}{2000 x + 5 - 2000 x + 5} = \lim_{x \to 0^{+}} \frac{{2 x}^{2}}{10} = 0

\lim_{x \to 0^{-}} f (x) = \lim_{x \to 0^{-}} \frac{x^{2} + 1 - (1 - x^{2})}{5 - 2000 x - (2000 x + 5)} = \lim_{x \to 0^{-}} \frac{{2 x}^{2}}{- 4000 x}

= - \lim_{x \to 0^{-}} \frac{x}{2000} = 0