If `f(x)={{:((cos^2x-sin^2x-1)/(sqrt(x^2+1)-1),if x ne 0),(k,if x=0):}`,is continuous at `x=0`
then find `k'.

Explanation

If `f(x)={{:((cos^2x-sin^2x-1)/(sqrt(x^2+1)-1),if x ne 0),(k,if x=0):}`,is continuous at `x=0`
then we have to seek out the value of `k`.
If f(x) is continuous at x=0, then
`lim_(x->0)f(x)=f(0)`
`lim_(x->0)(cos^2x-sin^2x-1)/(sqrt(x^2+1)-1)=k`
`lim_(x->0)((1-sin^2x)-sin^2x-1)/(sqrt(x^2+1)-1)=k`
`lim_(x->0)(-2sin^2x)/(sqrt(x^2+1)-1)=k`
`lim_(x->0)(-2sin^2x(sqrt(x^2+1)+1))/((sqrt(x^2+1)-1)(sqrt(x^2+1)+1))=k`
`=>-2lim_(x->0)(sin^2x(sqrt(x^2+1)+1))/ x^2 =k`
`=>-2lim_(x->0)((sinx)/x)lim_(x->0)(sqrt(x^2+1)+1)=k`
`=>k=-2xx1xx(1+1)=-4`