InterviewSolution
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InterviewSolution
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दर्शाइये कि फलन f(x), जहाँ `f(x)={{:((sin x)/(x)+cos x",",x gt 0),(2",",x = 0),((4(1-sqrt(1-x)))/(x)",",x lt 0):}` बिन्दु 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)," "[because x = 0 + h gt 0]` `=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]` `=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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