lim_(x→−2) (((√(x^2 +2x+4))−x^2 +2)/(x^5 +32)) no hospital


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Question Number 132483 by abdullahquwatan last updated on 14/Feb/21

$\lim_{x \to - 2} \frac{\sqrt{x^{2} + 2 x + 4} - x^{2} + 2}{x^{5} + 32} no hospital$
Commented by EDWIN88 last updated on 14/Feb/21

$what hospital ?$
Commented by abdullahquwatan last updated on 14/Feb/21

$l^{'} hopital theorem$
	Answered by EDWIN88 last updated on 14/Feb/21

	$let x + 2 = t \land t \to 0$ $\lim_{t \to 0} \frac{\sqrt{{(t - 2)}^{2} + 2 (t - 2) + 4} + 2 - {(t - 2)}^{2}}{{(t - 2)}^{5} + 32}$ $\lim_{t \to 0} \frac{\sqrt{t^{2} - 2 t + 4} - (t^{2} - 4 t + 2)}{{(t - 2)}^{5} + 32} =$ $\lim_{t \to 0} \frac{(t^{2} - 2 t + 4) - {(t^{2} - 4 t + 2)}^{2}}{{(t - 2)}^{5} + 32} \times \lim_{t \to 0} \frac{1}{\sqrt{t^{2} - 2 t + 4} + t^{2} - 4 t + 2}$ $= \frac{1}{4} \times \lim_{t \to 0} \frac{t^{2} - 2 t + 4 - (t^{4} + {4 t}^{2} + 4 - {8 t}^{3} + {4 t}^{2} - 16 t)}{t^{5} - {10 t}^{4} + {20 t}^{3} - {80 t}^{2} + 80 t}$ $= \frac{1}{4} \times \lim_{t \to 0} \frac{- t^{4} + {8 t}^{3} - {7 t}^{2} + 14 t}{t^{5} - {10 t}^{4} + {20 t}^{3} - {80 t}^{2} + 80 t}$ $= \frac{1}{4} \times \frac{14}{80} = \frac{7}{160}$
	Commented by abdullahquwatan last updated on 14/Feb/21

	$t h a n k y o u s o m u c h$
	Answered by bemath last updated on 14/Feb/21

	$If by L^{'} H \ddot{o p i t a l} rule$ $\lim_{x \to - 2} \frac{\frac{x + 1}{\sqrt{x^{2} + 2 x + 4}} - 2 x}{{5 x}^{4}} =$ $\frac{\frac{- 1}{\sqrt{4}} + 4}{80} = \frac{- \frac{1}{2} + 4}{80} = \frac{- 1 + 8}{160} = \frac{7}{160}$
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