यदि प्रदत फलन बिन्दु x = 0 पर संतत हो, तो a, b व c के मान ज्ञात कीजिए -
`f(x)={{:((sin(a+1)x+sin x)/(x)",","यदि",x lt 0),(" "c",","यदि",x = 0),((sqrt(x+bx^(2))-sqrt(x))/(bx^(3//2))",","यदि",x gt 0):}`

Explanation

(i) `f(0)=c`
(ii) R.H.L. `=underset(x rarr 0^(+))(lim)f(x)=underset(h rarr 0)(lim)f(0+h)`
`=underset(h rarr 0)(lim)(sqrt(h+bh^(2))-sqrt(h))/(bh^(3//2))`
`=underset(h rarr 0)(lim)(sqrt(h(1+bh))-sqrt(h))/(bh^(3//2))`
`=underset(h rarr 0)(lim)(sqrt(h)[sqrt((1+bh))-1])/(b sqrt(h)h)`
`=underset(h rarr 0)(lim)(sqrt(1+bh)-1)/(bh)xx(sqrt(1+bh)+1)/(sqrt(1+bh)+1)`, [परिमेयीकरण द्वारा]
`=underset(h rarr 0)(lim)((1+bh)-1)/(bh(sqrt(1+bh)+1))`
`=underset(h rarr 0)(lim)(bh)/(bh(sqrt(1+bh)+1))`
`=underset(h rarr 0)(lim)(1)/(sqrt(1+bh)+1)`
`rArr" ""R.H.L."=(1)/(1+1)=(1)/(2)`
(iii) L.H.L. `=underset(h rarr 0^(-))(lim)f(x)=underset(h rarr 0)(lim)f(0-h)`
`=underset(h rarr 0)(lim)(sin[(a+1)(0-h)]+sin(0-h))/((0-h))`
`=underset(h rarr 0)(lim)(sin[-(a+1)h]+sin(-h))/(-h)`
`=underset(h rarr 0)(lim)(-sin(a+1)h-sin h)/(-h)" "[because sin(-theta)=-sin theta]`
`=underset(h rarr 0)(lim)(sin(a+1)h + sin h)/(h)`
`=underset(h rarr 0)(lim)(sin(a+1)h)/(h)+underset(h rarr 0)(lim)(sin h)/(h)`
`=(a+1)underset(h rarr 0)(lim)(sin(a+1)h)/((a+1)h)+1`
`rArr" ""L.H.L."=(a+1)xx 1 + 1 = a + 2`
चूँकि फलन f(x) बिंदु x = 0 पर संतत है ।
`therefore" ""R.H.L.=L.H.L." = f(0)`
`rArr" "(1)/(2)=a+2=c`
`rArr" "a+2=(1)/(2)"और" c = (1)/(2)`
`rArr" "a=(1)/(2)-"2 और c"=(1)/(2)`
`rArr" "a=(-3)/(2)"और c"=(1)/(2)`
अत: `a=-(3)/(2),c=(1)/(2)` और b कोई भी वास्तविक मान ग्रहण कर सकता है ।