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If `x_(1)` and `x_(2)` are the real and distinct roots of `ax^(2)+bx+c=0` then prove that `lim_(xtox1) (1+sin(ax^(2)+bx+c))^((1)/(x-x_(1)))=e^(a(x_(1)-x_(2))).`

A

does not exist

B

1

C

`oo`

D

`(1)/(2)`

Text Solution

Verified by Experts

`ax^(2)+bx+c=a(x-x_(1))(x-x_(2))`
`underset(xtox1)lim(1+sin(ax^(2)+bx+c))^((1)/(x-x_(1)))" "(1^(oo))" form")`
`=e^(underset(xtox_(1))lim(sin(a(x-x_(1))(x-x_(2))))/((x-x_(1))))`
`=e^(underset(xtox_(1))lim(sin(a(x-x_(1)).(x-x_(2))))/(a(x-x_(1))(x-x_(2))).a(x-x_(2)))`
`=e^(a(x_(1)-x_(2)))`
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