InterviewSolution
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InterviewSolution
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निम्नलिखित फलनों की सांतत्यता की जाँच x = 0 पर कीजिए - (a) `f(x) = {{:((1-cos x)/(x^(2))",",x ne 0),((1)/(2)",",x = 0):}` (b) `f(x) = {{:((sin ax)/(sin bx)",",x ne 0),((a)/(b)",",x = 0):}` |
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Answer» (a) यहाँ `f(x)={{:((1-cos x)/(x^(2))",",x ne 0),(" "(1)/(2)",",x = 0):}` (i) `f(0) = (1)/(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)(1-cos(0+h))/((0+h)^(2))` `=underset(h rarr 0)(lim)(1-cos h)/(h^(2))` `=underset(h rarr 0)(lim)(1-(1-2 "sin"^(2) (h)/(2)))/(h^(2))" "[because cos x = 1 -2 "sin"^(2)(x)/(2)]` `=underset(h rarr 0)(lim)(2"sin"^(2)(h)/(2))/(h^(2))` `=2 underset(h rarr 0)(lim)(("sin"(h)/(2))/((h)/(2)))^(2)xx (1)/(4)` `=2 xx (1)^(2)xx(1)/(4)," "[because underset(theta rarr 0)(lim)(sin theta)/(theta)=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)(1-cos(0-h))/((0-h)^(2))` `=underset(h rarr 0)(lim)(1-cos h)/(h^(2))" "[because cos (-theta)=cos theta]` `=underset(h rarr 0)(lim)(2"sin"^(2)(h)/(2))/(h^(2))` `=2 underset(h rarr 0)(lim)(("sin"(h)/(2))/((h)/(2)))^(2)xx(1)/(4)` `=(1)/(2)` `therefore" "underset(x rarr 0^(+))(lim)f(x)=underset(x rarr 0^(-))(lim)f(x)=f(0)` अत: `f(x), x = 0` पर संतत है । (b) यहाँ `f(x)={{:((sin ax)/(sin bx)",",x ne 0),((a)/(b)",",x = 0):}` (i) `f(0) = (a)/(b)` (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 a(0+h))/(sin b(0+h))` `=underset(h rarr 0)(lim)(sin ah)/(sin bh)` `=underset(h rarr 0)(lim)[(sin ah)/(ah)xxahxx(bh)/(sinbh)xx(1)/(bh)]` `=(a)/(b)[underset(h rarr 0)(lim)(sin ah)/(ah)xx underset(h rarr 0)(lim)(1)/(((sin bh)/(bh)))]` `=(a)/(b)xx 1 xx 1," "[because h rarr 0 rArr ah rarr "0 और bh" rarr 0]` `=(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)(sin a(0-h))/(sin b(0-h))` `=underset(h rarr 0)(lim)(sin(-ah))/(sin(-bh))` `= underset(h rarr 0)(lim)(sin ah)/(sin bh)` `=(a)/(b)` `therefore" "underset(x rarr 0^(+))(lim)f(x)=underset(x rarr 0^(-))(lim)f(x)=f(0)=(a)/(b)` अत: `f(x), x = 0` पर संतत है । |
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