if-lim-x-0-ae-x-bsin-x-ce-x-xsin-x-2-what-is-a-b-c-

Question Number 80983 by jagoll last updated on 08/Feb/20

i f \lim_{x \to 0} \frac{{a e}^{x} - b \sin x + {c e}^{- x}}{x \sin x} = 2

w h a t i s a + b + c ?

Commented by jagoll last updated on 09/Feb/20

i {c a n}^{'} t s o l v e d t h i s e q u a t i o n .

w h a t {i t}^{'} s w r o n g ?

Commented by jagoll last updated on 09/Feb/20

s t e p (1) l i m i t m u s t b e \frac{0}{0}

\Rightarrow a + c = 0 \Rightarrow c = - a

s t e p (2) \lim_{x \to 0} \frac{{a e}^{x} - b \sin x - {a e}^{- x}}{x \sin x} = 2

L \hat{} h o p i t a l r u l e

\lim_{x \to 0} \frac{{a e}^{x} - b \cos x + {a e}^{- x}}{\sin x + x \cos x} = 2

s i n c e d e n u m e r a t o r = 0,

\Rightarrow 2 a - b = 0 \Rightarrow b = 2 a

L \hat{} h o p i t a l a g a i n

\lim_{x \to 0} \frac{{a e}^{x} + b \sin x - {a e}^{- x}}{2 \cos x - x \sin x} = 2

\Rightarrow \frac{0}{2} = 2 \Rightarrow ?

Commented by abdomathmax last updated on 09/Feb/20

\Rightarrow {l i m}_{x \to 0} \frac{{a e}^{x} - b s i n x + c e^{- x}}{x s i n x} - 2 = 0 \Rightarrow

{l i m}_{x \to 0} \frac{a e^{x} - b s i n x + {c e}^{- x} - 2 x s i n x}{x s i n x} = 0 \Rightarrow

{l i m}_{x \to 0} u (x) == {l i m}_{x \to 0} u^{^{'}} (x) = 0 a n d

{l i m}_{x \to 0} v (x) = {l i m}_{x \to 0} v^{^{'}} (x) = 0 w i t h

u (x) = {a e}^{x} - b s i n x + {c e}^{- x} - 2 x s i n x \Rightarrow

u (0) = 0 \Rightarrow a + c = 0 \Rightarrow c = - a

u^{^{'}} (x) = {a e}^{x} - b c o s x - {c e}^{- x} - 2 s i n x - 2 x c o s x

u^{^{'}} (0) = 0 \Rightarrow a - b - c = 0 \Rightarrow 2 a - b = 0 \Rightarrow b = 2 a

\Rightarrow a + b + c = 2 (b + c) = 2 (2 a - a) = 2 a

i f w e g e t a = 1 \Rightarrow a + b + c = 2

Commented by john santu last updated on 09/Feb/20

y o u a r e r i g h t s i r

Commented by jagoll last updated on 09/Feb/20

h o w t o g e t a = 1 ?

Commented by jagoll last updated on 09/Feb/20

i f w e g e t a = 1, i t m e a n f o r a = 6

a + b + c = 12 . t h a t m e a n s t h e

s o l u t i o n f r o m a + b + c i s n o t

o n e .

Commented by mr W last updated on 09/Feb/20

h a v e y o u c h e c k e d s i r ?

i {d o n}^{'} t t h i n k t h e a n s w e r i s c o r r e c t .

e x a m p l e : a = 1, b = 2, c = - 1 .

\lim_{x \to 0} \frac{e^{x} - 2 \sin x - e^{- x}}{x \sin x}

= \lim_{x \to 0} \frac{e^{x} - 2 \cos x + e^{- x}}{\sin x + x \cos x}

= \lim_{x \to 0} \frac{e^{x} + 2 \sin x - e^{- x}}{2 \cos x - x \sin x}

= \frac{0}{2} = 0 \neq 2

m y a n s w e r i s :

t h e r e e x i s t n o v a l u e s f o r a, b, c s u c h

t h a t \lim_{x \to 0} \frac{{a e}^{x} - b \sin x + {c e}^{- x}}{x \sin x} = 2 .

Commented by jagoll last updated on 09/Feb/20

y e s s i r . i t h i n k i t q u e s t i o n n o t r i g h t

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