Question

please answer the question please​

Answer 1

Answer:

solution to the question given above

Answer 2

$\\ \lim _{x →0} [\frac{ \sqrt{7 + x} - \sqrt{7 + {x}^{2} } }{ \sqrt{7 - {x}^{2} } - \sqrt{ 7 - x} } ]≡ \frac{0}{0}$

Using L'Hôspital's rule of Limits

$\\ \lim_{x→0} \frac{ \frac{1}{2 \sqrt{7 + x} } - \frac{2x}{ 2\sqrt{7 + {x}^{2} } } }{ - \frac{2x}{2 \sqrt{7 - {x}^{2} } } + \frac{1}{ 2\sqrt{7 - x} } }$

$\\ \lim _{x →0} \frac{( \sqrt{7 + {x}^{2} } - 2x \sqrt{7 + x} ) \sqrt{(7 - {x}^{2} )(7 - x)} }{( - 2x \sqrt{7 - x} + \sqrt{7 - {x}^{2} } ) \sqrt{(7 + x)(7 + {x}^{2} ) } }$

$\\ \lim _{x →0} \frac{ \sqrt{343 - 49x - 7 {x}^{4} + {x}^{5} } - 2x \sqrt{343 - 56 {x}^{2} + {x}^{4} } }{( \sqrt{7 - {x}^{2} - 2x \sqrt{7 - x} } ) \sqrt{49 + 7x + 7 {x}^{2} + {x}^{3} } }$

$\\ \implies \lim _{x →0} \frac{ \sqrt{7 + x} - \sqrt{7 + {x}^{2} } }{ \sqrt{7 - {x}^{2} } - \sqrt{ 7 - x} } → (1 )= \frac{ \sqrt{343} }{ \sqrt{7} \sqrt{49} }$