Show that the `underset(xto2)lim((sqrt(1-cos{2(x-2)}))/(x-2))` doesnot exist.

Options

A. `sqrt(2)`
B. `-sqrt(2)`
C. `2`
D. none of these

Answer

D

Explanation

We have
`lim_(xto2)sqrt(1-cos 2(x-2))/(x-2)=lim_(xto2) (sqrt1-|sin 2(x-2)|)/(x-2)`
Now,
`lim_(xto2^-)sqrt(1-cos 2(x-2))/(x-2)=lim_(xto2^-) (sqrt1-|sin 2(x-2)|)/(x-2)`
` =-sqrt(2) lim_(xto2) (sin(x-2))/(x-2)[because x lt 2rArr x-2lt 0 rArr sin (x-2) lt 0rArr |sin (x-2)|=- sin (x-2)]`
` =-sqrt(2)`
and,
`lim_(xto2^+)sqrt(1-cos 2(x-2))/(x-2)=lim_(xto2^+) (|sqrt2-sin (x-2)|)/(x-2)`
`=sqrt(2)lim_(xto2) (sin(x-2))/(x-2)=sqrt(2) [because x gt rArr x-2 gt0 rArr sin (x-2)gt 0 rArr |sin (x-2)|=sin (x-2)]`
`therefore lim_(xto2^-) sqrt(1-cos 2(x-2))/(x-2)ne lim_(xto2^+) sqrt(1-cos 2(x-2))/(x-2)`
`therefore lim_(xto2) sqrt(1-cos 2(x-2))/(x-2)` does not exist.