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Question

text  Lt  end text subscript x not stretchy rightwards arrow straight infinity end subscript left parenthesis square root of x plus 1 end root minus square root of x right parenthesis

  1. 1
  2. -1
  3. 0
  4. Infty

Hint:

In this question, we have to find the value for that first we will rationalize and then use the L' Hospital theorem and later put the value of x.

The correct answer is: 0


    text  Lt  end text subscript x not stretchy rightwards arrow straight infinity end subscript left parenthesis square root of x plus 1 end root minus square root of x right parenthesis
equals text  Lt  end text subscript x not stretchy rightwards arrow straight infinity end subscript left parenthesis square root of x plus 1 end root minus square root of x right parenthesis cross times fraction numerator left parenthesis square root of x plus 1 end root plus square root of x right parenthesis over denominator left parenthesis square root of x plus 1 end root plus square root of x right parenthesis end fraction
equals straight c fraction numerator x plus 1 minus x over denominator left parenthesis square root of x plus 1 end root plus square root of x right parenthesis end fraction
text = Lt  end text subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator 1 over denominator square root of x open parentheses square root of 1 plus begin display style 1 over x end style end root text +1 end text close parentheses end fraction
equals space 1 over infinity cross times fraction numerator 1 over denominator square root of 1 plus begin display style 1 over infinity end style end root plus 1 end fraction
equals 0 cross times 1 over 1
equals 0.

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