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Question

L t subscript left parenthesis x rightwards arrow 1 right parenthesis invisible function application left parenthesis left parenthesis 2 x minus 3 right parenthesis left parenthesis √ x minus 1 right parenthesis right parenthesis divided by left parenthesis 2 x squared plus x minus 3 right parenthesis equals

  1. space 1 divided by 10
  2. left parenthesis negative 1 right parenthesis divided by 10
  3. 1 divided by 5
  4. left parenthesis negative 1 right parenthesis divided by 5

hintHint:

We have to solve the given question of limit. We have to try to simplify the limit so we don't get indefinite form i.e /0/0 or infinity or infinity by infinity. We will assume the function to be f(x).

The correct answer is: left parenthesis negative 1 right parenthesis divided by 10


    We have to find the limit.
    limit as x rightwards arrow 1 of f open parentheses x close parentheses equals limit as x rightwards arrow 1 of fraction numerator left parenthesis 2 x space minus space 3 right parenthesis over denominator left parenthesis square root of x minus 1 right parenthesis left parenthesis 2 x squared plus x space minus space 3 right parenthesis 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 1 comma space w e space w i l l space g e t space d e n o m i n a t o r space t o space b e space z e r o.
space space space space space space space space space space space space
W e space w i l l space f a c t o r i s e
space left parenthesis 2 x squared plus x space minus space 3 right parenthesis space equals space left parenthesis 2 x squared space minus 2 x space plus space 3 x space minus space 3 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 equals space 2 x left parenthesis x space minus space 1 right parenthesis space plus space 3 left parenthesis x space minus space 1 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 equals space left parenthesis 2 x space plus space 3 right parenthesis space left parenthesis space x space minus 1 right parenthesis
    limit as x rightwards arrow 1 of f open parentheses x close parentheses equals limit as x rightwards arrow 1 of fraction numerator open parentheses 2 x minus 3 close parentheses left parenthesis square root of x minus 1 right parenthesis over denominator open parentheses 2 x plus 3 close parentheses left parenthesis x space minus space 1 right parenthesis end fraction
space space space space space space space space space space space space space space equals space limit as x rightwards arrow 1 of fraction numerator left parenthesis 2 x space minus space 3 right parenthesis open parentheses square root of x minus 1 close parentheses over denominator left parenthesis 2 x space plus space 3 right parenthesis left parenthesis x space minus space 1 right parenthesis end fraction cross times fraction numerator square root of x plus 1 over denominator square root of x plus 1 end fraction space space space space... left curly bracket space R a t i o n a l i z a t i o n right curly bracket
space space space space space space space space space space space space space equals limit as x rightwards arrow 1 of fraction numerator left parenthesis 2 x space minus space 3 right parenthesis left parenthesis x space minus space 1 right parenthesis over denominator left parenthesis 2 x space plus space 3 right parenthesis space left parenthesis x space minus space 1 right parenthesis left parenthesis square root of x plus 1 right parenthesis end fraction space space space space space space... left curly bracket space a squared minus b squared equals left parenthesis a space plus space b right parenthesis left parenthesis a space minus b right parenthesis
space space space space space space space space space space space space equals limit as x rightwards arrow 1 of fraction numerator left parenthesis 2 x space minus 3 right parenthesis over denominator left parenthesis 2 x space plus 3 right parenthesis left parenthesis square root of x plus 1 right parenthesis end fraction
space space space space space space space space space space space space equals fraction numerator left square bracket 2 left parenthesis 1 right parenthesis space minus space 3 right square bracket over denominator left square bracket 2 left parenthesis 1 right parenthesis space plus space 3 right square bracket left parenthesis square root of 1 plus 1 right parenthesis end fraction
space space space space space space space space space space space space equals fraction numerator 2 space minus 3 over denominator 5 left parenthesis 2 right parenthesis end fraction
space space space space space space space space space space space space equals fraction numerator negative 1 over denominator 10 end fractionThis is the final answer.

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

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