Maths-
General
Easy

Question

If L t subscript left parenthesis x rightwards arrow 0 right parenthesis invisible function application left parenthesis l o g invisible function application left parenthesis 3 plus x right parenthesis minus l o g invisible function application left parenthesis 3 minus x right parenthesis right parenthesis divided by x equals k space t h e n M space k equals

  1. 0
  2. negative 1 divided by 3
  3. 2 divided by 3
  4. left parenthesis negative 2 right parenthesis divided by 3

hintHint:

We are given a function. We have to find it's limit.

The correct answer is: 2 divided by 3


    We will denote the function by f(x)
    limit as x rightwards arrow 0 of f open parentheses x close parentheses equals limit as x rightwards arrow 0 of fraction numerator log left parenthesis 3 space plus space x right parenthesis space minus space log left parenthesis 3 space minus space x right parenthesis over denominator x end fraction
space space space space space space space space
    Now, we know that loga - logb = log open parentheses a over b close parentheses
    Consider the numerator log(3 + x) - log(3 - x) = log open parentheses fraction numerator 3 plus x over denominator 3 minus x end fraction close parentheses
    Now inside the bracket, we will divide the numerator and denominator by 3
    log(3 + x) - log(3 - x ) =              log left parenthesis 3 space plus space x right parenthesis space minus log left parenthesis 3 space minus x right parenthesis equals log left parenthesis fraction numerator begin display style fraction numerator 3 space plus space x over denominator 3 end fraction end style over denominator begin display style fraction numerator 3 minus x over denominator 3 end fraction end style end fraction right parenthesis
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals log open parentheses fraction numerator 1 space plus space begin display style 1 third end style x over denominator 1 space minus begin display style 1 third end style x end fraction close parentheses
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space log open parentheses 1 space plus space 1 third x close parentheses minus log open parentheses 1 space minus space 1 third x close parentheses
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals log open parentheses 1 space plus space 1 third x close parentheses minus log open parentheses 1 space plus left parenthesis negative 1 third right parenthesis x close parenthesesSubstituting the value.
    limit as x rightwards arrow 0 of f open parentheses x close parentheses equals limit as x rightwards arrow 0 of fraction numerator log open parentheses 1 space plus begin display style 1 third end style x close parentheses over denominator x end fraction minus log fraction numerator open parentheses 1 space plus left parenthesis begin display style fraction numerator negative 1 over denominator 3 end fraction end style right parenthesis x close parentheses over denominator x end fraction
space space space space space space space space space space space space space space space equals space 1 third minus open parentheses fraction numerator negative 1 over denominator 3 end fraction close parentheses
space space space space space space space space space space space space space space space equals fraction numerator 1 space plus 1 over denominator 3 end fraction
space space space space space space space space space space space space space space space equals 2 over 3
... left curly bracket space u sin g space f o r m u l a space limit as x rightwards arrow 0 of fraction numerator log left parenthesis 1 space plus space k x right parenthesis over denominator x end fraction equals space k space right curly bracket
    This is the final answer.

    For such questions, we should know different formulas.

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