`lim_(x rarr 0) (1-sqrt(1+x))/(x)` किसके बराबर है ?

Options

A. `1//2`
B. `-1//2`
C. 1
D. -1

Answer

B

Explanation

प्रथम विधि
`underset(xrarr0)lim(1-sqrt(1+x))/(x)`
`=underset(xrarr0)lim(1-sqrt(1+x))/(x)xx(1+sqrt(1+x))/(1+sqrt(1+x))` [ परिमेयीकरण करने पर ]
`=underset(xrarr0)lim(1-[sqrt((1+x))]^(2))/(x(1+sqrt(1+x)))`
`=underset(xrarr0)lim(1-(1+x))/(x(1+sqrt(1+x)))`
`=underset(xrarr0)lim(-x)/(x(+sqrt(1+x)))`
`=underset(xrarr0)lim(-1)/(1+sqrt(1+x))`
`=(-1)/(1+sqrt(1+0))=(-1)/(1+1)=(-1)/(2)`
द्वितीय विधि
`underset(xrarr0)lim(1-sqrt(1+x))/(x)` [`0/0` रूप]
L' हॉस्पिटल नियम से,
`=underset(xrarr0)lim(0-(1)/(2sqrt(1+x)))/(1)=(-1)/(2sqrt(1+0))=(-1)/(2)`