InterviewSolution
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InterviewSolution
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यदि `f(x)={{:((1-cos 4 x)/(x^(2))",","जब",x lt 0),(" "a",","जब",x = 0),(((sqrt(x)))/((16+sqrt(x))-4))",","जब",x gt 0):}` और f बिन्दु x = 0 पर संतत है, तो k का मान ज्ञात कीजिए । |
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Answer» (i) `f(0) = a` (ii) R.H.L. `=underset(x rarr 0^(+))(lim)f(x)=underset(h rarr 0)(lim)f(0+h)` `=underset(h rarr 0)(lim)(sqrt((0+h)))/((sqrt(16+sqrt((0+h))))-4)," "[because x = 0 + h gt 0]` `=underset(h rarr 0)(lim)(sqrt(h))/(sqrt(16+sqrt(h)))-4` `=underset(h rarr 0)(lim)(sqrt(h))/((sqrt(16+sqrt(h)))-4)xx(sqrt(16+sqrt(h))+4)/((sqrt(16+sqrt(h)))+4)` `=underset(h rarr 0)(lim)(sqrt(h)[sqrt(16+sqrt(h))+4])/(sqrt(h))` `=underset(h rarr 0)(lim)(sqrt(h)[sqrt(16+sqrt(h))+h])/(sqrt(h))` `=underset(h rarr 0)(lim)[sqrt(16+sqrt(h))+4]` `=sqrt(16+0)+4` `=4 + 4 = 8` (iii) L.H.L. `underset(x rarr 0^(-))(lim)f(x)=underset(h rarr 0)(lim)f(0-h)` `=underset(h rarr 0)(lim)(1-cos 4(0-h))/((0-h)^(2))," "[because x = 0 - h lt 0]` `=underset(h rarr 0)(lim)(1-cos 4h)/(h^(2))` `=underset(h rarr 0)(lim)(1-(1-2 sin^(2)2h))/(h^(2))," "[because sin 2 theta = 1 - 2 sin^(2)theta]` `=underset(h rarr 0)(lim)(2 sin^(2)2h)/(h^(2))` `=2 underset(h rarr 0)(lim)((sin 2h)/(2h))^(2)xx 4` `= 2 xx 1 xx 4 = 8` चूँकि दिया गया फलन x = 0 पर संतत है । `therefore" ""R.H.L.=L.H.L."=f(0)` `rArr" "a = 8` |
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