Find the value of the constant `k` so that the function given below is continuous at `x=0` . `f(x)={(1-coskx)/(xsinx),\ \ \ x!=0\ \ \ \ \ \ \ \ and \ \ \ \ 1/2 \ \ \ \ when \ \ x=0`

Options

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

Answer

A

Explanation

Since f(x) is continuous at x = 0, so

`lim_(x->0)f(x)=f(0)=1/2`

`lim_(x->0)f(x)=1/2`

`lim_(x->0)((1-coskx)/(xsinx))=1/2`

`lim_(x->0)((ksinkx))/(sinx+xcosx)=1/2`

`lim_(x->0)((ksinkx)/x)/((sinx)/x+xcosx)=1/2`

`lim_(x->0)((k^2sinkx)/(kx))/((sinx)/x+cosx)=1/2`

`(k^2)/2=(1/2)`

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