Home
Class 12
MATHS
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_(xrarr -1)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`
Promotional Banner

Topper's Solved these Questions

  • CONIC SECTIONS

    CENGAGE PUBLICATION|Exercise All Questions|102 Videos

  • COORDINATE SYSTEM

    CENGAGE PUBLICATION|Exercise Multiple Correct Answers Type|2 Videos

Similar Questions

Explore conceptually related problems

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 (1) it is continuous everywhere except when x=-1 ,(2) lim_(x->oo)h(x)=oo and (3) lim_(x->-1)h(x)=1/2 then the value of h(1)

Let f(x)=sinx,g(x)=x^2 and h(x) = log x. If F(x)=(h(f(g(x)))) , then F'(x) is:

If f(x)=sinx, g(x)=x^2 and h(x)=log x, find h[g{f(x)}].

If the function f(x)=x^(3)+e^((x)/(2)) and g(x)=f^(-1)(x) , then the value of g'(1) is

Let f(x)=a^(x)(a gt 0) be written as f(x)=g(x)+h(x) where g(x) is an even function and h(x) is an odd function. Then the value of g(x+y)+g(x-y) is -

f(x)= x^2+xg'(1)+g''(2) and g(x)=f(1)x^2+xg'(1)+f''(x) . The domain of the function sqrt(f(x)/g(x)) is-

If the function f(x) satisfies lim_(xrarr1)(f(x)-2)/(x^(2)-1)=pi , evaluate lim_(xrarr1)f(x) .

If f(x)=(1)/(x^2) , show that f(x+h)-f(x-h)=-(4xh)/((x^2-h^2)^2) .

If f(x)=(x+1)/(x-1)a n dg(x)=1/(x-2),t h e n discuss the continuity of f(x),g(x),a n dfog(x)dot

Let f be a differentiable function such that f(1) = 2 and f'(x) = f (x) for all x in R . If h(x)=f(f(x)) , then h'(1) is equal to

CENGAGE PUBLICATION-CONTINUITY AND DIFFERENTIABILITY-All Questions
  1. Let f(x)=[(1-sinpix)/(1+cos2pix), x<1/2 and p , x=1/2 and sqrt(2x-1)...

    Text Solution

    |

  2. For which of the following functions f(0) exists such that f(x) is con...

    Text Solution

    |

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

    Text Solution

    |

  4. Let f(x) = (1 - x)^2 sin^2 x + x^2 for all x ∈ R, and let g(x) = ∫((2...

    Text Solution

    |

  5. If f(x)={sin(pi/2)(x-[x]),x<5 5(b-1),x=5(a b^2|x^2-11 x+24|)/(x-3),x >...

    Text Solution

    |

  6. The function f(x) is discontinuous only at x=0 such that f^2(x)=1AAx i...

    Text Solution

    |

  7. f(x)={(x^2+e^(1/(2-x)))^(-1)k ,x=2, x!=2 is continuous from right at t...

    Text Solution

    |

  8. If the function f(x)=((128 a+a x)^(1//8)-2)/((32+b x)^(1//5)-2) is con...

    Text Solution

    |

  9. If f(x)={(1-cos(1-cos x/2))/(2^m x^n)1x=0,x!=0 is continous at x=0 the...

    Text Solution

    |

  10. f is a continous function in [a, b]; g is a continuous function in [b,...

    Text Solution

    |

  11. If the function f(x) defined as f(x) defined as f(x)={3,x=0(1+(a x+b x...

    Text Solution

    |

  12. Let f(x)={((alpha cotx)/x+beta/x^2 ,, 0<|x|lt=1),(1/3 ,, x=0):}. If f...

    Text Solution

    |

  13. Let f(x)={2/(1+x^2),xi sr a t ion a l b ,xi sr a t ion a l has exactly...

    Text Solution

    |

  14. Let f(x)=[t a n x[cot x]],x [pi/(12),pi/(12)] , (where [.] denotes the...

    Text Solution

    |

  15. Number of points of discontinuity of f(x)=[sin^(-1)x]-[x] in its domai...

    Text Solution

    |

  16. If g(x)=(lim)(mvecoo)(x^mf(x)+h(x)+3)/(2x^m+4x+1) when x!=1a n dg(1)=e...

    Text Solution

    |

  17. The number of points of discontinuity of fx)=[2x^2]-{2x2}^2 (where [] ...

    Text Solution

    |

  18. If f(x)={(|x|-3 when x < 1), and (|x-2|+a , when x >= 1) & g(x)={2...

    Text Solution

    |

  19. The function f(x)=(x^3)/8-s inpix+4in[-4,4] does not take the value -4...

    Text Solution

    |

  20. Let f(x) be continuous functions f: RvecR satisfying f(0)=1a n df(2x)-...

    Text Solution

    |