QJEEM19Mae5e7: JEE Main 2019 Mathematics - Class 12 JEEM19Mae5e7 Solution

Question

Verified Solution • 2025–26 Session moderate

\mathop {\lim }\limits_{y \to 0} {{\sqrt {1 + \sqrt {1 + {y^4}} } - \sqrt 2 } \over {{y^4}}}

Accepted Answer

\mathop {\lim }\limits_{y \to 0} {{\sqrt {1 + \sqrt {1 + {y^4}} } - \sqrt 2 } \over {{y^4}}}

If you put y = 0 at

{{\sqrt {1 + \sqrt {1 + {y^4}} } - \sqrt 2 } \over {{y^4}}}

it is in

{0 \over 0}

form. So we can use L' Hospital's Rule.

=

\mathop {\lim }\limits_{y \to 0} {{{1 \over {2\sqrt {1 + \sqrt {1 + {y^4}} } }} \times \left( {{1 \over {2\sqrt {1 + {y^4}} }}} \right) \times 4{y^3}} \over {4{y^3}}}

=

\mathop {\lim }\limits_{y \to 0} {1 \over {2\sqrt {1 + \sqrt {1 + {y^4}} } }} \times {1 \over {2\sqrt {1 + {y^4}} }}

=

{1 \over {4\sqrt 2 }}

Step-by-Step Verified Solution