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    Maths-
    General
    Easy

    Question

    L t left parenthesis x rightwards arrow 0 right parenthesis invisible function application left parenthesis √ left parenthesis left parenthesis 1 plus x plus x squared right parenthesis right parenthesis minus 1 right parenthesis divided by x

    1. 1 divided by 2
    2. 1
    3. 2
    4. 0

    hintHint:

    We are given a function. We have to find the limit of the function.

    The correct answer is: 1 divided by 2

      The given function is
      f open parentheses x close parentheses equals fraction numerator square root of 1 plus x plus x squared end root space minus space 1 over denominator x end fraction
      We have to find the limit of the given function.
      limit as x rightwards arrow 0 of f left parenthesis x right parenthesis space equals space limit as x rightwards arrow 0 of fraction numerator square root of x plus x squared plus 1 end root minus 1 over denominator x end fraction space space space space space space space I f space w e space s u b s t i t u t e space x space equals space 0 comma space w e space g e t space z e r o space i n space d e n o m i n a t o r. S o comma space w e space n e e d space t o space a j u s t space t h e space f u n c t i o n. space space W e space w i l l space u s e space r a t i o n a l i z a t i o n space m e t h o d space t o space s o l v e space f u r t h e r space space space space space space space space space space space space
      limit as x rightwards arrow 0 of f left parenthesis x right parenthesis space equals limit as x rightwards arrow 0 of fraction numerator square root of x space plus space x squared plus 1 space end root minus 1 over denominator x end fraction space cross times space fraction numerator square root of x space plus space x squared plus space 1 end root space plus 1 over denominator square root of x plus x squared plus 1 end root space plus space 1 end fraction space space space space space space space space space space space space space space equals limit as x rightwards arrow 0 of fraction numerator left parenthesis x space plus space x squared plus space 1 right parenthesis space minus 1 over denominator x left parenthesis square root of x space plus space x squared plus 1 end root plus space 1 right parenthesis end fraction space space space space space space... left curly bracket a squared minus b squared equals left parenthesis a space minus b right parenthesis left parenthesis a space plus space b right parenthesis right curly bracket space space space space space space space space space space space space space equals space limit as x rightwards arrow 0 of fraction numerator x space plus space x squared over denominator x left parenthesis square root of x space plus space x squared plus 1 end root plus 1 right parenthesis end fraction space space space space space space space space space space space space equals limit as x rightwards arrow 0 of space fraction numerator x left parenthesis 1 space plus space x right parenthesis over denominator x left parenthesis square root of x space plus space x squared plus 1 end root space plus 1 end fraction space space space space space space space space space space space space equals limit as x rightwards arrow 0 of fraction numerator 1 space plus space x over denominator square root of x space plus space x squared plus space 1 end root plus 1 end fraction space space space space space space space space space space space space equals fraction numerator 1 space plus space 0 over denominator square root of 0 space plus space 0 space plus space 1 end root plus space 1 end fraction space space space space space space space space space space space space space equals space fraction numerator 1 over denominator 1 space plus space 1 end fraction space space space space space space space space space space space equals 1 half space space space space space space space space space space space space space space spaceThis is the final answer.

      For such questions, we should know different formulas of limits.

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