k के किस मान के लिए फलन ` f(x)= {:{((x^(2) -16)/(x-4),x!= 4),(k,x=4):}` x=4 पर संतत है ।

Explanation

` :' ` x = 4 पर f (x)का मान k है ।
` :. F (4) = k `
x = 4 पर ` R.H.L = lim _( x to 4 + 0 ) f ( x ) `
` = lim _( h to 0 ) f ( 4 + h ) `
` = lim -(h to 0 ) ((4 +h)^(2) - 16 )/(4 + h - 4 ) = lim _(h to 0 ) ( 16 + h^(2) + 8h - 16 )/h`
` = lim _(h to 0 ) (h ( h + 8 ))/h = lim _(h to 0 ) ( h + 8 ) " " [ :' h != 0 ] `
`=8`
इसी प्रकार ` L.H.L = lim _( x to 4 - 0) f(x)`
` = lim _( h to 0 ) f ( 4- h ) `
` = lim _( h to 0 ) (( 4-h)^(2) - 16 )/( 4-h-4) = lim _ ( h to 0) ( 16 + h^(2) - 8h - 16)/(-h)`
` = lim _( h to 0) ( -h + 8) " " [ :' h !0 ]`
=8
दिया है कि फलन` x = 4` पर संतत है ।
` L.H.L = R.H . L = f ( 4) `
` 8 = 8 = k `
अतः ` k = 8 `