InterviewSolution
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InterviewSolution
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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)):}` |
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Answer» (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` |
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