1.

Let `f(x)={{:(,sum_(r=0)^(x^(2)[(1)/(|x|)])r,x ne 0),(,k,x=0):}` where [.] denotes the greatest integer function. The value of k for which is continuous at x=0, isA. 1B. 2C. 4D. `(1)/(2)`

Answer» Correct Answer - A
Clearly, f(x) is an even function and is given by `f(x)={{:(,(x^(2))/(2)[(1)/(|x|)] ([(1)/(|x|)]+1),x ne 0),(,k,x=0):}`
If f is continous at x=0, then
`underset(x to 0)lim f(x)=f(0)`
`underset(x to 0)lim (x^(2))/(2)[(1)/(|x|)] ([(1)/(|x|)]+1)=k`
`Rightarrow underset(x to 0)lim ([(1)/(|x|)] ([(1)/(|x|)]+1))/((1)/(|x|^(2)))=2k`
`Rightarrow underset( y to oo)lim ([y]([y]+1))/(y^(2))-2k,"where "y=(1)/(|x|)`
`Rightarrow underset( y to oo)lim ([y])/(y) (([y])/(y)+(1)/(y))=2k`
`Rightarrow 1(1+0)=2k" "[therefore underset(x to oo)lim (x)/([x])=1]`
`Rightarrow k=1`


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