यदि `f(x)={{:(x^(3)+3,if,xne0),(1,if,x=0):}x=0` पर फलन की संतत तथा अवकलनीयता कि जाँच कीजिये |

Explanation

x = 0 पर संतता
`f (0) = 4" " `…(1)
` LHL = underset (x to 0-0) lim f(x) `
` = underset ( h to 0 ) lim f (0-h) = underset (h to 0)lim (-h) `
` = underset ( h to 0 ) lim {(-h)^(3)+ 3 } `
` = underset ( h to 0 ) lim ( -h^(3)-+ 3 ) " "`...(2)
` RHL = underset ( x to 0 + 0 ) lim f ( x ) = underset ( h to 0) lim f ( 0+h)`
` = underset ( h to 0 ) lim f ( h ) `
` = underset ( h to 0 ) lim ( h^(3) + 3 ) = 3 " "` ...(3)
` :' LHL = RHL != f ( 0 ) `
` :. f (x ) , x = 0 ` पर संतत नहीं है ।
`x=0` पर अवकलनीयता
` Lf '(0) = underset ( x to 0)lim ( f(x) - f (0))/(x-0)`
` = underset ( h to 0 ) lim ( f ( 0-h) - f ( 0 ))/ (0-h-0)`
` = underset (h to 0 ) lim ( f (-h ) - f ( 0 ) ) /(-h) `
` = underset (h to 0 ) lim ((-h)^(3) + 3 - 4 ) /(-h) `
` underset ( h to 0 ) lim (( -h^(3)-1)/(-h))`
`= underset (h to 0 ) lim (h^(2) + 1 / h )= oo " " ` ... (4)
` Rf' ( 0 ) = underset ( h to 0 ) lim (f(x)- (0))/(x-0)`
` = underset ( h to 0 ) lim (f(0+h) - f ( 0 ) )/ (0+h-0) `
` = underset ( h to 0 ) lim (f(h)- f ( 0 ) )/h `
` = underset ( h to 0 ) lim ( h^(3) - 3-4)/h `
` = underset ( h to 1) lim ((h^(3)-1)/h)`
` = underset (h to 0 ) lim ( h^(2) - 1/h ) = - oo " "` ...(5)
` :' Lf' ( 0 ) != Rf'(0)`
` :. f ( x ) , x = 0 ` पर अवकलनीय नहीं है ।