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Let f(x)=x^(3)-x^(2)-3x-1, g(x)=(x+1)a a...

Let `f(x)=x^(3)-x^(2)-3x-1, g(x)=(x+1)a and h(x)=(f(x))/(g(x))` where h is a rational function such that
(i) It is continuous everywhere except when `x=-1`,
(ii) `lim_(xrarroo)h(x)=(1)/(2)`.
The value of h(1) is

A

`1//2`

B

`1//4`

C

`-1//2`

D

1

Text Solution

Verified by Experts

The correct Answer is:
C
`h(x)=f(x)/(g(x))`
`rArr" "h(x)=(x^(3)-x^(2)-3x-1)/((x+1)a)" (i)"`
Now, `" "underset(xrarr-1)(lim)h(x)=(1)/(2)`
`rArr" "underset(xrarr-1)(lim)(x^(3)-x^(2)-3x-1)/((x+1)a)=(1)/(2)`
`rArr" "(2)/(a)=(1)/(2)rArr a=4`
`therefore" "h(x)=(x^(3)-x^(2)-3x-1)/(4(x+1))`
`therefore" "h(1)=-1//2`
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