InterviewSolution
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InterviewSolution
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`f(x)={{:((1+2x)^(2)",","जब",x ne 2),(e^(2)",","जब",x = 0):}x = 0` पर सांतत्य की जाँच कीजिए । |
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Answer» (i) `f(0) = e^(2)` (ii) `f(0+0)=underset(h rarr 0)(lim)f(0+h)=underset(h rarr 0)(lim)[1 + 2(0+h)]^((1)/(0+h))` `=underset(h rarr 0)(lim)[1+2h]^((1)/(h))` `=underset(h rarr 0)(lim)[1+2h]^((1)/(2h)xx2)` माना 2h = t, जब `h rarr 0`, तब `t rarr 0` `therefore" "f(0+0)=underset(t rarr 0)(lim)[1+t]^((1)/(t)xx2)` `rArr" "f(0+0)=e^(2)," "[because underset(t rarr 0)(lim)(1+x)^((1)/(x))=e]` (iii) `f(0-0)=underset(h rarr 0)(lim)f(0-h)underset(h rarr 0)(lim)[1 + 2(0-h)]^((1)/(0-h))` `=underset(h rarr 0)(lim)[1-2h]^((1)/(-h))` `=underset(h rarr 0)(lim)[1-2h]^((1)/(-2h)xx2)` माना `-2h = t` जब `h rarr 0`, तब `t rarr 0` `therefore" "f(0-0)=underset(t rarr 0)(lim)[1+t]^((1)/(t)xx2)," "[because underset(x rarr 0)(lim)(1+x)^((1)/(x))=e]` `rArr" "f(0-0)=e^(2)` `because f(0 + 0)=f(0-0)=f(0)` अत: दिया गया फलन x = 0 पर संतत है । |
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