Home
Class 12
MATHS
If f is an odd function and if Lim(x to ...

If f is an odd function and if `Lim_(x to 0) f(x)` exists , prove that this limit must be zero .

A

0

B

-1

C

1

D

non-existent

Text Solution

Verified by Experts

The correct Answer is:
A
It is given that `lim_(xto0)f(x)` exists
`therefore lim_(xto0^-)f(x)=lim_(xto0^+)f(x)`
`rArr lim_(hto0)f(0-h)=lim_(hto0)f(0+h)`
`rArr lim_(xto0)f(-h)=lim_(hto0)f(h)`
`rArr-lim_(hto0)f(h)=lim_(hto0)f(h)[because f(x) "is good"therefore f(-h)=-f(h)]`
`rArr 2lim_(hto0)f(h)=0rArr lim_(hto0)f(h)=0rArr lim_(xto0)f(x)=0`
Promotional Banner

Topper's Solved these Questions

  • LIMITS

    OBJECTIVE RD SHARMA ENGLISH|Exercise Section I - Solved Mcqs|101 Videos

  • LIMITS

    OBJECTIVE RD SHARMA ENGLISH|Exercise Section II - Assertion Reason Type|5 Videos

  • INVERSE TRIGONOMETRIC FUNCTIONS

    OBJECTIVE RD SHARMA ENGLISH|Exercise Chapter Test|30 Videos

  • MATHEMATICAL INDUCTION

    OBJECTIVE RD SHARMA ENGLISH|Exercise Exercise|30 Videos

Similar Questions

Explore conceptually related problems

If f is an odd function and f(0) is defined, must f(0)=?

If f(x) = (x^(2)-9)/(x-3) , find it Lim_(x to 3) f(x) exists .

Let f be an even function and f'(x) exists, then f'(0) is

For the function f(x) = 2 . Find lim_(x to 1) f(x)

If f(x) is continuous function in [0,2pi] and f(0)=f(2 pi ), then prove that there exists a point c in (0,pi) such that f(x)=f(x+pi).

Prove that Lim_(x to 0) (|x|)/x , x != 0 does not exist .

If lim_(x->a)(f(x)/(g(x))) exists, then

Statement I The function f(x) = int_(0)^(x) sqrt(1+t^(2) dt ) is an odd function and STATEMENT 2 : g(x)=f'(x) is an even function , then f(x) is an odd function.

If f is an even function, then prove that lim_(x->0^-) f(x) = lim_(x->0^+) f(x)

If f is an even function , then prove that Lim_( xto 0^(-)) f(x) = Lim_(x to 0^(+)) f(x) , whenever they exist .

OBJECTIVE RD SHARMA ENGLISH-LIMITS-Chapter Test
  1. If f is an odd function and if Lim(x to 0) f(x) exists , prove that th...

    Text Solution

    |

  2. Let f(x)={(x^(2),x epsilonZ),((d(x^(2)-4))/(2-x),x !inZ):} the set of ...

    Text Solution

    |

  3. If Sn=sum(k=1)^n ak and lim(n->oo)an=a , then lim(n->oo)(S(n+1)-Sn)/sq...

    Text Solution

    |

  4. If a1=1a n da(n+1)=(4+3an)/(3+2an),ngeq1,a n dif("lim")(nvecoo)an=a , ...

    Text Solution

    |

  5. If x(1)=3 and x(n+1)=sqrt(2+x(n))" ",nge1, then underset(ntooo)limx(n)...

    Text Solution

    |

  6. The value of underset(xrarr0)(lim)(sqrt(1-cosx^(2)))/(1-cos x) is

    Text Solution

    |

  7. Evaluate underset(ntooo)limncos((pi)/(4n))sin((pi)/(4n)).

    Text Solution

    |

  8. Evaluate ("lim")(n→oo){cos(x/2)cos(x/4)cos(x/8)... cos(x/(2^n))}

    Text Solution

    |

  9. If f(x) is the integral of (2 sin x - sin 2x )/(x ^ 3 ) , w...

    Text Solution

    |

  10. Evaluate: ("lim")(xvec0)x^m(logx)^n ,m , n in Ndot

    Text Solution

    |

  11. The value of lim(xrarroo) (logx)/(x^n), n gt 0, is

    Text Solution

    |

  12. underset(xtoa)lim(log(x-a))/(log(e^(x)-e^(a)))

    Text Solution

    |

  13. Let < an > be a sequence such that lim(x->oo)an=0. Then lim(n->oo)(a1...

    Text Solution

    |

  14. If f(a)=2,f^(prime)(a)=1,g(a)=-1,g^(prime)(a)=2, then the value of ("l...

    Text Solution

    |

  15. If f(9)=9,f^(prime)(9)=4,t h e n("lim")(nvecoo)(sqrt(f(x)-3))/(sqrt(x-...

    Text Solution

    |

  16. A(i)=(x-a(i))/(|x-a(i)|),i=1,2,...,n," and "a(1)lta(2)lta(3)lt...lta(n...

    Text Solution

    |

  17. lim(x -> oo) x^n / e^x = 0, (n is an integer) for

    Text Solution

    |

  18. lim(xrarr0) (x)/(tan^-1x) is equal to

    Text Solution

    |

  19. If f(x) =x , x<0 and f(x)=1 , x = 0, and f(x)=x^2,x>0 then lim(x->0) ...

    Text Solution

    |

  20. Evaluate the following limits : Lim(x to oo) sqrt(((x+sin x)/(x- cos...

    Text Solution

    |

  21. Evaluate: ("lim")(xvecoo)(1+1/(a+b x))^(c+dx),w h e r ea , b , c ,a n ...

    Text Solution

    |