Question

solve this ???I want to know the answer​

Answer 1

SOLUTION

TO EVALUATE

$\displaystyle \sf{\lim_{x \to 0 \: } \: \frac{ \sqrt{1 + {x}^{2} } - \sqrt{1 + x} }{ \sqrt{1 + {x}^{3} } - \sqrt{1 + x} } }$

EVALUATION

$\displaystyle \sf{\lim_{x \to 0 \: } \: \frac{ \sqrt{1 + {x}^{2} } - \sqrt{1 + x} }{ \sqrt{1 + {x}^{3} } - \sqrt{1 + x} } }$

$\displaystyle \sf{ = \lim_{x \to 0 \: } \: \frac{( \sqrt{1 + {x}^{2} } - \sqrt{1 + x} )( \sqrt{1 + {x}^{2} } + \sqrt{1 + x} ) (\sqrt{1 + {x}^{3} } + \sqrt{1 + x} )}{ (\sqrt{1 + {x}^{3} } + \sqrt{1 + x} )(\sqrt{1 + {x}^{3} } - \sqrt{1 + x} )( \sqrt{1 + {x}^{2} } + \sqrt{1 + x} )} }$

$\displaystyle \sf{ = \lim_{x \to 0 \: } \: \frac{( 1 + {x}^{2} - 1 - x ) (\sqrt{1 + {x}^{3} } + \sqrt{1 + x} )}{ (1 + {x}^{3} - 1 - x )( \sqrt{1 + {x}^{2} } + \sqrt{1 + x} )} }$

$\displaystyle \sf{ = \lim_{x \to 0 \: } \: \frac{( {x}^{2} - x ) (\sqrt{1 + {x}^{3} } + \sqrt{1 + x} )}{ ( {x}^{3} - x )( \sqrt{1 + {x}^{2} } + \sqrt{1 + x} )} }$

$\displaystyle \sf{ = \lim_{x \to 0 \: } \: \frac{x( x - 1 ) (\sqrt{1 + {x}^{3} } + \sqrt{1 + x} )}{ x( {x}^{2} - 1 )( \sqrt{1 + {x}^{2} } + \sqrt{1 + x} )} }$

$\displaystyle \sf{ = \lim_{x \to 0 \: } \: \frac{x( x - 1 ) (\sqrt{1 + {x}^{3} } + \sqrt{1 + x} )}{ x( x + 1)(x - 1)( \sqrt{1 + {x}^{2} } + \sqrt{1 + x} )} }$

$\displaystyle \sf{ = \lim_{x \to 0 \: } \: \frac{x(\sqrt{1 + {x}^{3} } + \sqrt{1 + x} )}{ (x + 1)( \sqrt{1 + {x}^{2} } + \sqrt{1 + x} )} }$

$\displaystyle \sf{ = \: \frac{0(\sqrt{1 + {0}^{3} } + \sqrt{1 + 0} )}{ (0 + 1)( \sqrt{1 + {0}^{2} } + \sqrt{1 + 0} )} }$

$\displaystyle \sf{ = \frac{0 \times 2}{1 \times 2} }$

$\displaystyle \sf{ = 0 }$

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