k का मान ज्ञात कीजिए यदि फलन f(x) बिन्दु `x = (pi)/(2)` पर संतत है, जहाँ `f(x)={{:((k cos x)/(pi - 2x)",","यदि",x ne (pi)/(2)),(3",","यदि",x = (pi)/(2)):}`

Explanation

(i) `f((pi)/(2))=3`
(ii) R.H.L. `=underset(x rarr (pi^(+))/(2))(lim)f(x)=underset(h rarr 0)(lim)f((pi)/(2)+h)`
`= underset(h rarr 0)(lim)(k cos((pi)/(2)+h))/(pi-2((pi)/(2)+h))," "[because x = (pi)/(2)+h ne (pi)/(2)]`
`=underset(h rarr 0)(lim)(k(-sin h))/(pi-(pi + 2h))," "[because cos ((pi)/(2)+theta)=-sin theta]`
`=underset(h rarr 0)(lim)(k sin h)/(2h)`
`= (k)/(2)underset(h rarr 0)(lim)(sin h)/(h)`
`=(k)/(2) xx 1=(k)/(2)," "[because underset(theta rarr 0)(lim)(sin theta)/(theta)=1]`
(iii) L.H.L. `=underset(x rarr (pi^(-))/(2))(lim)f(x)=underset(h rarr 0)(lim)f((pi)/(2)-h)`
`=underset(h rarr 0)(lim)(k cos ((pi)/(2)-h))/(pi-2((pi)/(2)-h))`
`=underset(h rarr 0)(lim)(k sin h)/(pi-(pi+2h))`
`=underset(h rarr 0)(lim)(k sin h)/(2h)`
`=(k)/(2)underset(h rarr 0)(lim)(sin h)/(h)=(k)/(2)xx 1 = (k)/(2)`
चूँकि फलन f(x) बिन्दु `x = (pi)/(2)` पर संतत है ।
`therefore" ""R.H.L.=L.H.L."=f((pi)/(2))`
`rArr" "(k)/(2)=3`
`rArr" "k = 6`