यदि फलन ` f(x) = {:{((x^(2) -4)/(x-2),x != 2),(k,x=2):}` x=2 पर संतत है तो k का मान ज्ञात कीजिए ।

Options

A. `k=2`
B. `k=3`
C. `k=4`
D. `k=5`

Answer

C

Explanation

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