फलन `f(t)` के सांतत्यता की जाँच `t = (pi)/(2)` पर कीजिए -
`f(t) = {{:((cos t)/((pi)/(2)-t)",",t ne (pi)/(2)),(1",",t = (pi)/(2)):}`

Options

A. सतत होगा
B. सतत नहीं होगा
C. कुछ नहीं कहा जा सकता
D. इनमे से कोई नहीं

Answer

A

Explanation

(i) `f((pi)/(2))=1`
(ii) R.H.L. `=underset(t rarr (pi^(+))/(2))(lim)f(t)=underset(h rarr 0)(lim)f((pi)/(2)+h)`
`=underset(h rarr 0)(lim)(cos((pi)/(2)+h))/((pi)/(2)-((pi)/(2)+h))`
`=underset(h rarr 0)(lim)(-sin h)/(-h)`
`=underset(h rarr 0)(lim)(sin h)/(h)=1`
(iii) L.H.L. `=underset(t rarr (pi^(-))/(2))(lim)f(t)=underset(h rarr 0)(lim)f((pi)/(2)-h)`
`=underset(h rarr 0)(lim)(cos((pi)/(2)-h))/((pi)/(2)-((pi)/(2)-h))`
`= underset(h rarr 0)(lim)(sin h)/(h)`
= 1.
`therefore" "underset(t rarr (pi^(+))/(2))(lim)f(t)=underset(t rarr (pi^(-))/(2))(lim)f(t)=1`
अत: `f(t), t=(pi)/(2)` पर संतत है ।