lim-x-2-ax-2-bx-6-x-2-x-2-10-find-a-b-

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 AST 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 \dots (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 \dots (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 \dots (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

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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

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Commented by esmaeil last updated on 03/Feb/24

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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 \dots (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