Calculer-lim-x-2-x-2-x-2-6-x-2-

Question Number 216316 by a.lgnaoui last updated on 03/Feb/25

Calculer

\lim_{x \to - 2} \frac{x^{2} + x - 2}{\sqrt{6 + x} - 2}

Answered by A5T last updated on 03/Feb/25

\frac{x^{2} + x - 2}{\sqrt{6 + x} - 2} = \frac{(x + 2) (x - 1) (\sqrt{6 + x} + 2)}{2 + x}

= (x - 1) (\sqrt{6 + x} + 2)

\Rightarrow \lim_{x \to - 2} (\frac{x^{2} + x - 2}{\sqrt{6 + x} - 2}) = - 3 \times 4 = - 12

Answered by MrGaster last updated on 04/Feb/25

\lim_{x \to 2} \frac{(x - 1) (x + 2)}{\sqrt{6 + x} - 2}

= \lim_{x \to - 2} \frac{(x - 1) (x + 2) (\sqrt{6 + x} + 2)}{(\sqrt{6 + x} - 2) (\sqrt{6 + x} + 2)}

= \lim_{x \to - 2} \frac{(x - 1) (x - 2) (\sqrt{6 + x} - 2)}{6 + x - 4}

= \lim_{x \to - 2} \frac{(x - 1) (x - 2) (\sqrt{6 - x} + 2)}{x - 2}

= \lim_{x \to - 2} (x - 1) (\sqrt{6 + x} + 2)

= (- 2 - 1) (\sqrt{6 - 2} + 2)

= (- 3) (\sqrt{4} + 2)

= (- 3) (2 + 2)

= (- 3) (4)

= - 12

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