Find the value of the constant `k` so that the given function is continuous at the indicated point: `f(x)={{:((x-1)tan((pix)/2),\ \ \ if\ x!=1),(k ,\ \ \ if\ x=1):}` at `x=1`

Explanation

Given


`f(x)={c}(x-1) tan frac{pi x}{2}, text { if } x neq 1 k, text { if } x=1` If `f(x)` is continuous at `x=1`, then `lim _{x -> 1} f(x)=f(1)`


`Rightarrow lim _{x -> 1}(x-1) tan frac{pi x}{2}=k`


Putting `x-1=y`, we get


`lim _{y -> 0} y tan frac{pi(y+1)}{2}=k`


`Rightarrow lim _{y -> 0} y tan (frac{text { piy }}{2}+frac{pi}{2})=k`


`Rightarrow lim _{y -> 0} y tan (frac{pi}{2}+frac{p i y}{2})=k`


`Rightarrow-lim _{y -> 0} y cot (frac{p i y}{2})=k`


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