Home
Class 11
MATHS
If f is an even function, prove that li...

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

Text Solution

Verified by Experts

Let f be an even function Then,
`underset(xto0^(-))limf(x)=underset(hto0)limf(0-h)=underset(hto0)limf(-h)`
`=underset(hto0)limf(h)" "[becausef"being even,"f(-h=f(h)]=underset(hto0)limf(0+h)=underset(xto0^(+))limf(x).`
Hence, `underset(xto0^(-))limf(x)=underset(xto0^(+))limf(x).`
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

Evaluate lim_(xto0^(+)) x^(x)"and "lim_(xto0^(+))x^(x^(x))

Evaluate lim_(xto0^(+))(sinx)^(x)

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

Evaluate lim_(xto0^(+)) (1)/(x)cos^(-1)((sinx)/(x)) .

If f (x) is a thrice differentiable function such that lim _(xto0)(f (4x) -3 f(3x) +3f (2x) -f (x))/(x ^(3))=12 then the vlaue of f '''(0) equais to :

Evaluate lim_(xto0)|x|^(sinx)

What is lim_(xto0)(a^(x)-b^(x))/(x) ?

Evaluate lim_(xto0) (sinx-x)/(x^(3)).

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

    |