`lim_(x rarr 0) (x)/(sqrt(1-cos x))` किसके तुल्य है ?

Options

A. `sqrt(2)`
B. `-sqrt(2)`
C. `(1)/(sqrt(2))`
D. सीमा का अस्तित्व नहीं है

Answer

D

Explanation

`underset(hrarr0)lim(2-(1.5+h)-(2-1.5))/(h)`
`=underset(hrarr0)lim(0.5-h-0.5)/(h)`
`=underset(hrarr0)lim-(h)/(h)=-1`
बाईं पक्ष सीमा
`=f(0-0)=underset(hrarr0)lim(0-h)`
`=(1)/(sqrt(2))underset(hrarr0)lim(-h)/(|sin((-h)/(2))|)`
`=-(1)/(sqrt(2))underset(hrarr0)lim(h)/(|-sin(h/2)|)`
`-(1)/(sqrt(2))underset(hrarr0)lim(2.(h/2))/(sin.(h)/(2))=-(1)/(sqrt(2))xx2xx1 [:'underset(theta rarr0)lim(theta)/(sin theta)=1]`
`=-sqrt(2)`
दाईं पक्ष सीमा
`=f(0+0)=underset(hrarr0)limf(0+h)`
`=(1)/(sqrt(2))underset(hrarr0)lim(h)/(|sin.(h)/(2)|)=(1)/(sqrt(2))underset(hrarr0)lim(2(h/2))/(sin(h/2))`
`=(1)/(sqrt(2))xx2xx1 [:'underset(theta rarr0)lim(theta)/(sin theta)=1]`
`=sqrt(2)`
`:'` बाईं पक्ष सीमा `!=` दाईं पक्ष सीमा
अतः सीमा विद्यमान नहीं है ।