Maths-
General
Easy

Question

Lt subscript x not stretchy rightwards arrow straight infinity end subscript open parentheses fraction numerator x plus 6 over denominator x plus 1 end fraction close parentheses to the power of x plus 1 end exponent

  1. e to the power of 6
  2. e to the power of 4
  3. e
  4. e cubed

hintHint:

The initial form for the limit is indeterminate 1 to the power of infinity to the power of infinity  So, use the formula. In this question, we have to find value of Lt subscript x not stretchy rightwards arrow straight infinity end subscript open parentheses fraction numerator x plus 6 over denominator x plus 1 end fraction close parentheses to the power of x plus 1 end exponent.

The correct answer is: e cubed


    Lt subscript x not stretchy rightwards arrow straight infinity end subscript open parentheses fraction numerator x plus 6 over denominator x plus 1 end fraction close parentheses to the power of x plus 1 end exponent
    Lt subscript x not stretchy rightwards arrow straight infinity end subscript space open parentheses fraction numerator 1 plus begin display style 6 over x end style over denominator 1 plus begin display style 1 over x end style end fraction close parentheses to the power of x plus 1 end exponent space space space space left parenthesis W e space c a n space w r i t e space space left parenthesis x plus a right parenthesis space equals space x left parenthesis 1 plus a over x right parenthesis. space space space right parenthesis
    Lt subscript x not stretchy rightwards arrow straight infinity end subscript space open parentheses fraction numerator 1 plus begin display style 6 over x end style over denominator 1 plus begin display style 1 over x end style end fraction close parentheses to the power of x plus 1 end exponent equals space space open parentheses fraction numerator 1 plus begin display style a over infinity end style over denominator 1 plus begin display style b over infinity end style end fraction close parentheses to the power of infinity space equals 1 to the power of infinity
    bold 1 to the power of bold infinity bold space bold italic f bold italic o bold italic r bold italic m this is known as an indeterminate form, because it is unknown. One to the power infinity is unknown because infinity itself is endless. Take a look at some examples of indeterminate forms. To solve this limit we will use the following formula -
    Error converting from MathML to accessible text.
    Lt subscript x not stretchy rightwards arrow straight infinity end subscript space open parentheses fraction numerator x plus 6 over denominator x plus 1 end fraction close parentheses to the power of x plus 1 end exponent equals e to the power of left parenthesis space Lt subscript straight x not stretchy rightwards arrow straight infinity end subscript space stretchy left parenthesis fraction numerator straight x plus 6 over denominator straight x plus 1 end fraction minus 1 stretchy right parenthesis. left parenthesis straight x plus 1 right parenthesis space right parenthesis end exponent
e to the power of left parenthesis space Lt subscript straight x not stretchy rightwards arrow straight infinity end subscript space stretchy left parenthesis fraction numerator 6 minus 1 over denominator straight x plus 1 end fraction stretchy right parenthesis. left parenthesis straight x plus 1 right parenthesis space right parenthesis end exponent space equals e to the power of left parenthesis space Lt subscript straight x not stretchy rightwards arrow infinity end subscript space e to the power of left parenthesis 5 right parenthesis end exponent end exponent space equals space e to the power of 5

    The basic problem of this indeterminate form is to know from where f not stretchy left parenthesis x not stretchy right parenthesis tends to one (right or left) and what function reaches its limit more rapidly.

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