Let `f(x)={((cos^(2)x-sin^(2)x-1)/(sqrt(x^(2)+4)-2)"," , x ne 0),(a",", x =0):}`, then the value of a in order that f(x) may be continuous at x = 0 is

Options

A. `-8`
B. 8
C. `-4`
D. 4

Answer

A

Explanation

Given, `f(x) ={((cos^(2)x-sin^(2)x-1)/(sqrt(x^(2)+4)-2)",", x ne 0),(a",",x =0):}`
be continuous at x = 0, then
`lim_(x to 0)f(x)=lim_(x to0)(cos^(2)x-sin^(2)x-1)/(sqrt(x^(2)+4)-2)=lim_(x to 0)(cos2x-1)/(sqrt(x^(2)+4)-2)`
`=lim_(x to 0) (-2 sin 2x)/( (2x)/(2sqrt(x^(2)+4))) " " `[by L' Hospital's rule]
`=lim_(x to0) (-2 sin 2x sqrt(x^(2)+4))/(x)`
`=lim_(x to 0)[4cos 2x sqrt(x^(2)+4)+2 sin 2x(x)/(sqrt(x^(2)+4))]`[by L' Hospital's rule ]
`=-[4sqrt(4)]=-8`
Thus, `lim_(x to 0) f(x)=-8`
Since, function is continuous at x = 0.
`therefore f(0)=lim_(x to0)f(x) rArr a = -8`