Maths-
General
Easy

Question

text  Lt  end text subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator a subscript 0 plus a subscript 1 x to the power of 1 plus a subscript 2 x squared plus horizontal ellipsis plus a subscript n x to the power of n over denominator b subscript 0 plus b subscript 1 x to the power of 1 plus b subscript 2 x squared plus horizontal ellipsis comma plus b subscript m x to the power of n end fraction text  where  end text a subscript n greater than 0 comma b subscript m greater than 0 text  and  end text n greater than m

  1. 1
  2. 0
  3. does not exist

  4. straight infinity

hintHint:

In this question, we have to find value of text  Lt  end text subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator a subscript 0 plus a subscript 1 x to the power of 1 plus a subscript 2 x squared plus horizontal ellipsis plus a subscript n x to the power of n over denominator b subscript 0 plus b subscript 1 x to the power of 1 plus b subscript 2 x squared plus horizontal ellipsis comma plus b subscript m x to the power of n end fraction text  where  end text a subscript n greater than 0 comma b subscript m greater than 0 text  and  end text n greater than m.

The correct answer is: straight infinity


    text  Lt  end text subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator a subscript 0 plus a subscript 1 x to the power of 1 plus a subscript 2 x squared plus horizontal ellipsis plus a subscript n x to the power of n over denominator b subscript 0 plus b subscript 1 x to the power of 1 plus b subscript 2 x squared plus horizontal ellipsis comma plus b subscript m x to the power of n end fraction text  where  end text a subscript n greater than 0 comma b subscript m greater than 0 text  and  end text n greater than m
    Also, We can write same equation in another manner,
    text  Lt  end text subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator a subscript 0 plus a subscript 1 x to the power of 1 plus a subscript 2 x squared plus horizontal ellipsis plus a subscript n x to the power of n over denominator b subscript 0 plus b subscript 1 x to the power of 1 plus b subscript 2 x squared plus horizontal ellipsis comma plus b subscript m x to the power of n end fraction text  =  Lt  end text subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator begin display style a subscript 0 over x to the power of n end style plus begin display style a subscript 1 over x to the power of n minus 1 end exponent end style plus begin display style a subscript 2 over x to the power of n minus 2 end exponent end style x plus horizontal ellipsis plus a subscript n over denominator begin display style b subscript 0 over x to the power of n end style plus begin display style b subscript 1 over x to the power of n minus 1 end exponent end style x to the power of 1 plus begin display style b subscript 2 over x to the power of n minus 2 end exponent end style plus horizontal ellipsis plus b subscript m end fraction
    On substituting, We get
    space Lt space subscript x not stretchy rightwards arrow straight infinity end subscript fraction numerator begin display style a subscript 0 over x to the power of n plus a subscript 1 over x to the power of n minus 1 end exponent plus a subscript 2 over x to the power of n minus 2 end exponent x plus horizontal ellipsis plus a subscript n end style over denominator begin display style b subscript 0 over x to the power of n plus b subscript 1 over x to the power of n minus 1 end exponent x to the power of 1 plus b subscript 2 over x to the power of n minus 2 end exponent plus horizontal ellipsis plus b subscript m end style end fraction space equals space space Lt space subscript x not stretchy rightwards arrow straight infinity end subscript a subscript n over b subscript m
    space Lt space subscript x not stretchy rightwards arrow straight infinity end subscript a subscript n over b subscript m space equals a subscript n over b subscript m

    We can only apply the L’Hospital’s rule if the direct substitution returns an indeterminate form, that means 0 over 0 or infinity over infinity.

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