Maths-
General
Easy

Question

L t subscript left parenthesis x rightwards arrow infinity right parenthesis invisible function application left parenthesis left parenthesis x squared plus 5 x plus 3 right parenthesis divided by left parenthesis x squared plus x plus 2 right parenthesis right parenthesis to the power of x equals

  1. e to the power of 4
  2. e cubed
  3. e squared
  4. e to the power of t

hintHint:

In this question we are getting 1 to the power of infinity form. So, we will use the standard limits formula limit as x rightwards arrow infinity of left parenthesis f left parenthesis x right parenthesis to the power of g left parenthesis x right parenthesis end exponent equals space e to the power of limit as x rightwards arrow infinity of left parenthesis f left parenthesis x right parenthesis minus 1 right parenthesis g left parenthesis x right parenthesis end exponent.

The correct answer is: e to the power of 4


    In this question we have to find the limit of limit as x rightwards arrow infinity of open parentheses fraction numerator x squared plus 5 x plus 3 over denominator x squared plus x plus 2 end fraction close parentheses to the power of x
    Step1: Putting the value of limit.
    By putting the value of limit in the expression we are getting 1 to the power of infinity form.
    Step2: Using Standard limit.
    We know that limit as x rightwards arrow infinity of left parenthesis f left parenthesis x right parenthesis to the power of g left parenthesis x right parenthesis end exponent equals space e to the power of limit as x rightwards arrow infinity of left parenthesis f left parenthesis x right parenthesis minus 1 right parenthesis g left parenthesis x right parenthesis end exponent
    =>e to the power of limit as x rightwards arrow infinity of open parentheses fraction numerator x squared plus 5 x plus 3 over denominator x squared plus x plus 2 end fraction minus 1 close parentheses cross times x end exponent
    => e to the power of limit as x rightwards arrow infinity of open parentheses fraction numerator x squared plus 5 x plus 3 minus x squared minus x minus 2 over denominator x squared plus x plus 2 end fraction close parentheses x end exponent
    =>e to the power of limit as x rightwards arrow infinity of open parentheses fraction numerator 4 x plus 1 over denominator x squared plus x plus 2 end fraction close parentheses x end exponent
    => e to the power of limit as x rightwards arrow infinity of open parentheses fraction numerator 4 x squared plus x over denominator x squared plus x plus 2 end fraction close parentheses end exponent
    => e to the power of limit as x rightwards arrow infinity of open parentheses fraction numerator 4 plus begin display style 1 over x end style over denominator 1 plus begin display style 1 over x end style plus begin display style 2 over x squared end style end fraction close parentheses end exponent
    By putting the value of limit in the expression we get,
    =>e to the power of 4
    Hence, the limit of the expression is e to the power of 4.

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