InterviewSolution
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InterviewSolution
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यदि फलन f(x) जहाँ `f(x)={{:((log(1+ax)-log(1-bx))/(x)",",x ne 0),(" "k",",x = 0):}x = 0` पर संतत है, तब k का मान ज्ञात कीजिए । |
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Answer» (i) `f(0) = k` (ii) R.H.L. `=underset(x rarr 0^(+))(lim)f(x) = underset(h rarr 0)(lim)f(0+h)` `=underset(h rarr 0)(lim)(log(1+a(0+h))-log(1-b(0+h)))/((0+h))` `underset(h rarr 0)(lim)(log(1+ah)-log(1-bh))/(h)` `=underset(h rarr 0)(lim)(log(1+ah))/(h)-underset(h rarr 0)(lim)(log(1-bh))/(h)` `=a xx underset(h rarr 0)(lim)(log (1+ah))/(ah)-(-b)underset(h rarr 0)(lim)(log (1-bh))/((-b)h)` `= a xx 1 - (-b) xx 1," "[because underset(x rarr 0)(lim)(log (1+x))/(x)=1]` `rArr" ""R.H.L." = a + b` (iii) L.H.L. `underset(x rarr 0^(-))(lim)f(x) = underset(h rarr 0)(lim)f(0-h)` `underset(h rarr 0)(lim)(log(1+a(0-h))-log(1-b(0-h)))/((0-h))` `=underset(h rarr 0)(lim)(log(1-ah)-log(1+bh))/(-h)` `=underset(h rarr 0)(lim)(log(1-ah))/(-h)+underset(h rarr 0)(lim)(log(1+bh))/(h)` `= a xx underset(h rarr 0)(lim)(log(1-ah))/((-a)h)+b xx underset(h rarr 0)(lim)(log(1+x))/(bh)` `rArr" ""L.H.L." = a xx 1 + b xx 1 = a + b, " "[because underset(x rarr 0)(lim)(log(1+x))/(x)=1]` चूँकि फलन f(x) बिन्दु x = 0 पर संतत है । `therefore" ""R.H.L.=L.H.L." = f(0)` `rArr" "a + b =k`. |
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