`k` का मान ज्ञात कीजिए यदि फलन `x = 0` पर संतत है, जहाँ
`f(x)={{:((1-cos 4x)/(8x^(2))if ,x ne 0),(" kif ,x = 0):}`

Options

A. `1`
B. `2`
C. `3`
D. `4`

Answer

A

Explanation

(i) `f(0)=k`
(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 4(0+h))/(8(0+h)^(2))" "[because x = 0 + h ne 0]`
`=underset(h rarr 0)(lim)(1-cos 4h)/(8h^(2))`
`=underset(h rarr 0)(lim)(2sin^(2)2h)/(8h^(2))`
`=underset(h rarr 0)(lim)((sin 2h)/(2h))^(2)`
`=1^(2)=1," "[because underset(theta rarr 0)(lim)(sin theta)/(theta)=1]`
(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 4(0-h))/(8(0-h)^(2))`
`=underset(h rarr 0)(lim)(1-cos 4h)/(8h^(2))," "[because cos (-theta)=cos theta]`
= 1
चूँकि फलन f(x) बिन्दु x = 0 पर संतत है ।
`therefore" ""R.H.L.=L.H.L."=f(0)`
`rArr" "k = 1`