lim_(x→2) ((ax^2 +bx+6)/(x^2 −x−2))=10 ; find a=? ∧ b=?


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Question Number 203980 by Davidtim last updated on 03/Feb/24

$\lim_{x \to 2} \frac{{a x}^{2} + b x + 6}{x^{2} - x - 2} = 10; f i n d a = ? \land b = ?$
	Answered by deleteduser1 last updated on 03/Feb/24

	$\frac{a (x^{2} - x - 2) + (b + a) x + 6 + 2 a}{x^{2} - x - 2} = 10$ $\Rightarrow \underset{x \to 2}{l i m} [a + \frac{(b + a) x + 6 + 2 a}{(x - 2) (x + 1)}] = 10$ $\Rightarrow \underset{x \to 2}{l i m} \frac{(b + a) x + 6 + 2 a}{(x - 2) (x + 1)} = 10 - a$ $2 (b + a) + 6 + 2 a = 0 \Rightarrow 2 a + b = - 3 . . . (i)$ $\Rightarrow \underset{x \to 2}{l i m} \frac{(b + a) x + 6 + 2 a}{(x - 2) (x + 1)} = \underset{x \to 2}{l i m} \frac{b + a}{2 x - 1} = \frac{b + a}{3} = 10 - a$ $\Rightarrow b + a = 30 - 3 a \Rightarrow 4 a + b = 30 . . . (i i)$ $(i) \land (i i) \Rightarrow a = 16.5 \Rightarrow b = - 36$
	Answered by mr W last updated on 03/Feb/24

	$\lim_{x \to 2} ({a x}^{2} + b x + 6) = a \times 2^{2} + b \times 2 + 6 = 0$ $\Rightarrow 2 a + b = - 3 . . (i)$ $\lim_{x \to 2} = \frac{{a x}^{2} + b x + 6}{x^{2} - x - 2} = \lim_{x \to 2} = \frac{2 a x + b}{2 x - 1} = \frac{2 a \times 2 + b}{2 \times 2 - 1} = 10$ $\Rightarrow 4 a + b = 30 . . . (i i)$ $(i i) - (i) :$ $\Rightarrow a = \frac{33}{2} = 16.5 ✓$ $\Rightarrow b = - 3 - 2 \times \frac{33}{2} = - 36 ✓$
	Commented by esmaeil last updated on 03/Feb/24

	$h i$ $w h y d o y o u u s e h o p i t a l ?$
	Commented by mr W last updated on 03/Feb/24

	$w h y n o t ? {i t}^{'} s t h e r e t o b e u s e d .$
	Commented by esmaeil last updated on 03/Feb/24

	${{a x}^{2} + b x + 6]}_{x = 2} \overset{? ?}{=} 0$
	Commented by mr W last updated on 03/Feb/24

	$s u c h t h a t l i m i t e x i s t s,$ ${{a x}^{2} + b x + 6]}_{x = 2} \overset{m u s t}{=} 0 . o t h e r w i s e$ $\lim_{x \to 2} \frac{{a x}^{2} + b x + 6}{x^{2} - x - 2} = \infty$
	Commented by esmaeil last updated on 03/Feb/24

	$\underset{―}{\underset{⏟}{⋚}}$
	Commented by Mathspace last updated on 03/Feb/24

	$s i r s m a e l h o s p i t a l i s g o o d i n t h i s c a s e . . .$
	Commented by esmaeil last updated on 03/Feb/24

	$o k .$ $t h a n k s .$ $w h i c h i s r i g h t ?$ $h o p i t a l o r h o s p i t a l . ?$
	Commented by mr W last updated on 03/Feb/24

	$L^{'} H o p i t a l i s r i g h t . b u t L^{'} H o s p i t a l$ $i s a l s o c o m m o n l y s p e l l e d .$
	Commented by siyathokoza last updated on 07/Feb/24

	$\lim_{x \to 2} ({a x}^{2} + b x + 6) = a \times 2^{2} + b \times 2 + 6 = 0$ $\Rightarrow 2 a + b = - 3 . . (i)$ $\lim_{x \to 2} = \frac{{a x}^{2} + b x + 6}{x^{2} - x - 2} = \lim_{x \to 2} = \frac{2 a x + b}{2 x - 1} = \frac{2 a \times 2 + b}{2 \times 2 - 1} = 10$ $\Rightarrow 4 a + b = 30 . . . (i i)$ $(i i) - (i) :$ $\Rightarrow a = \frac{33}{2} = 16.5 ✓$ $\Rightarrow b = - 3 - 2 \times \frac{33}{2} = - 36 ✓$
	Commented by mr W last updated on 07/Feb/24

	$d o e s i t m a k e a n y s e n s e t h a t y o u$ $p o s t a c o p y f r o m o t h e r {p e o p l e}^{'} s$ $w o r k i n g s ?$
	Answered by Rasheed.Sindhi last updated on 03/Feb/24

	$\lim_{x \to 2} \frac{{a x}^{2} + b x + 6}{x^{2} - x - 2} = 10; f i n d a = ? \land b = ?$ $x^{2} - x - 2 = (x - 2) (x + 1)$ ${a x}^{2} + b x + c = (x - 2) (a x - 3)$ $= {a x}^{2} - (3 + 2 a) x + 6$ $b = - (3 + 2 a)$ $\lim_{x \to 2} \frac{{a x}^{2} + b x + 6}{x^{2} - x - 2} = \lim_{x \to 2} \frac{a x - 3}{x + 1} = 10$ $\frac{a (2) - 3}{2 + 1} = 10$ $2 a = 33 \Rightarrow a = 33 / 2$ $b = - (3 + 2 (33 / 2) = - 36$
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