InterviewSolution
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InterviewSolution
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दर्शाइये कि फलन f(x) जहाँ `f(x)={{:((sin x)/(x)+cos x",",x ne 0),(" "2",",x = 0):} x = 0` पर संतत है । |
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Answer» (i) `f(0)=2` (ii) R.H.L. `=underset(x rarr 0^(+))(lim)f(x)=underset(h rarr 0)(lim)f(0+h)` `=underset(h rarr 0)(lim)[(sin (0+h))/(0+h)+cos(0+h)]` `=underset(h rarr 0)(lim)[(sin h)/(h)+cos h]` `=underset(h rarr 0)(lim)(sin h)/(h)+underset(h rarr 0)(lim)cos h` `= 1 + 1 = 2` (iii) L.H.L. `=underset(x rarr 0^(-))(lim)f(x)=underset(h rarr 0)(lim)f(0-h)` `=underset(h rarr 0)(lim)[(sin (0-h))/((0-h))+cos (0-h)]` `=underset(h rarr 0)(lim)[(sin(-h))/(-h)+cos(-h)]` `=underset(h rarr 0)(lim)[(-sin h)/(-h)+cos h]," "[because sin (-theta)=-sin theta,cos(-theta)=cos theta]` `=underset(h rarr 0)(lim)[(sin h)/(h)+cos h]` `=underset(h rarr 0)(lim)(sin h)/(h) + underset(h rarr 0)(lim)cos h` `= 1 + 1 = 2` `therefore" "underset(x rarr 0^(+))(lim)f(x)=underset(x rarr 0^(-))(lim)f(x)=f(0)=2` अत: फलन f(x) बिन्दु x = 0 पर संतत है । |
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