Home
Class 11
MATHS
If is an odd function and underset(xto0...

If is an odd function and `underset(xto0)f(x)` exists then prove that this limit must be 0.

Text Solution

Verified by Experts

Let f be an odd function and let `underset(xto0)lim f(x)` exist. Then
`underset(xto0^(+))limf(x)=underset(xto0^(-))limf(x)`
`impliesunderset(hto0)limf(0+h)=underset(hto0)limf(0+h)`
`impliesunderset(hto0)limf(h)=underset(hto0)limf(-h)=-underset(hto0)limf(h)[becausef "being odd", f(-h)=-f(h)]`
`implies2underset(hto0)limf(h)=0impliesunderset(hto0)limf(h)=0impliesf(x)=0.`
Hence, `underset(xto0) f(x)=0.`
Promotional Banner

Topper's Solved these Questions

  • LIMIT

    RS AGGARWAL|Exercise EXERCISE 27A|40 Videos

  • LIMIT

    RS AGGARWAL|Exercise EXERCISE 27B|72 Videos

  • LIMIT

    RS AGGARWAL|Exercise EXERCISE 27C|15 Videos

  • HYPERBOLA

    RS AGGARWAL|Exercise EXERCISE 24|23 Videos

  • LINEAR INEQUATIONS (IN ONE VARIABLE)

    RS AGGARWAL|Exercise EXERCISE 6B|12 Videos

Similar Questions

Explore conceptually related problems

If f is an odd function and if lim_(x rarr0)f(x) exists. Prove that this limit must be zero.

If f(0)=0" and "underset(xto0)("lim")(f(x))/(x) exists then this limits is equl to

Let f be an even function and f'(0) exists then find f'(0)

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

If |f(x)|lex^(2), then prove that lim_(xto0) (f(x))/(x)=0.

If f(x) is an odd function in R and f(x)={-x,0 then definition of f(x) in (-oo,0) is

underset(xto0)("lim")(int_(0)^(x)t tan (5t)dt)/x^3 is equal to :

If f(x)=(1)/(1+e^(-(1)/(x))),x!=0 and 0,x=0 then at x=0 (A) right hand limit of f(x) exists but not left-hand limit (B) limit of f(x) exists but not but not right-hand limit (C) both limits exists but are not equal (D) both limits exist and are equal

RS AGGARWAL-LIMIT-SOLVED EXAMPLES
  1. Evaluate lim(xrarr0)((x^(3)cotx)/(1-cosx)).

    Text Solution

    |

  2. lim(x->0)(sin3x+7x)/(4x+sin2x)

    Text Solution

    |

  3. Evaluate lim(x-> 0) (tan3x-2x)/(3x- sin^2 x)

    Text Solution

    |

  4. Evaluate lim(xrarr0)(xtan4x)/(1-cos4x).

    Text Solution

    |

  5. Evaluate lim(xrarr0)((1-cosxsqrt(cos2x)))/(x^(2)).

    Text Solution

    |

  6. Evaluate lim(xrarr(pi)/(4))((sinx-cosx))/((x-(pi)/(4))).

    Text Solution

    |

  7. Evaluate lim(xrarr(pi)/(2))(cosx)/(((pi)/(2)-x)).

    Text Solution

    |

  8. Evaluate lim(xrarr(pi)/(6))((sqrt3sinx-cosx))/((x-(pi)/(6))).

    Text Solution

    |

  9. Let f(x){:{(2x+3",",xle0),(3(x+1)",",xgt0.):} Find (i) lim(xrarr0)f(...

    Text Solution

    |

  10. Let f(x)={:{(x^(2)-1",",xle1),(-x^(2)-1",",xgt1.):} "Find"lim(xrarr1)f...

    Text Solution

    |

  11. Let f(x)={:{((|x|)/(x)",",xne0),(0",",x=0.):} Find lim(xrarr0)f(x).

    Text Solution

    |

  12. Suppose f(x)={(a+bx, x<1), (4, x=1), (b-ax, x>1):} and if lim(xrarr1) ...

    Text Solution

    |

  13. If f(x)={{:(|x|+1,xlt0),(0,x=0) ,(|x|-1,xgt0):} for what value (s) ...

    Text Solution

    |

  14. Let f(x)={{:(mx^(2)+n",",xlt0),(nx+m",",0lexle1),(nx^(3)+m",",xgt1):} ...

    Text Solution

    |

  15. Let a(1),a(2),...,a(n) be fixed real numbers and let f(x)=(x-a(1))(...

    Text Solution

    |

  16. Let f(x)={x}= greater integer less than or eqal to x. For any integer ...

    Text Solution

    |

  17. If is an odd function and underset(xto0)f(x) exists then prove tha...

    Text Solution

    |

  18. If f is an even function, prove that lim(xto0^(-)) (x)=lim(xto0^(+)) ...

    Text Solution

    |

  19. Show that lim(xto0^(-)) ((e^(1//x)-1)/(e^(1//x)+1)) does not exist.

    Text Solution

    |

  20. Show that ("lim")(x->0)x/(|x|) does not exist.

    Text Solution

    |