If the following function f(x) is continuous at x=0 , find the values of `a, b and c`. `f(x) ={ {:( (sqrt(x+bx^(2))-sqrt(x))/(bx^(3/2)) ," if " x gt 0),( " " c, " if " x =0),((sin (a+1)x+sin x)/(x) , " if " x lt 0):}`

If the following function f(x) is continuous at x=0 , find the values

1.	If the following function f(x) is continuous at x=0 , find the values of `a, b and c`. `f(x) ={ {:( (sqrt(x+bx^(2))-sqrt(x))/(bx^(3/2)) ," if " x gt 0),( " " c, " if " x =0),((sin (a+1)x+sin x)/(x) , " if " x lt 0):}`
Answer» we havef(0)=a ` lim_(x to 0-) f( 0-h)` `lim_(h to0) (sin (a +1) (-h) +sin(-h))/((-h)) = lim_( hto0)(-{sin (a+1) h +sin h})/(-h)` `lim_(h to o) (sin(a+1) +sinh)/h = lim_(hto0) (2sin (a/2 +1)h.cos(ah)/2)/h ` `2.lim_(h to0) {(sin (a/2=1)h)/((a/2+1)h).(a/2+1).cos""(ah)/2}` `2(a/2+1) .lim_(h to0) (sin (a/2=1)h)/((a/2+1)).lim_(hto0) cos""(ah)/2` ` (a + 2) xx 1 xx 1 = (a +2)` ` lim_(hto0) {([sqrt(h+bh^(2))-sqrth])/(bh^(3//2))xx([sqrt(h+bh^(2))+sqrth])/([sqrt(h +bh^(2))+sqrth])` ` lim_(hto0) ((h+bh^(2)-h))/(bh^(3//2) (sqrt(h+bh^(2))+sqrth)) =lim_(h to 0) (bh^(2))/(bh^(2)(sqrt(1+bh+1))` `lim_(hto0) 1/((sqrt(1+bh+1)))=1/2` since f (x) is continuous at x =0, we have `f(0) =lim_(xto0-)f(x) Rightarrow c=1/2 anda+2=1/2` `c = 1/2 and a = (-3)/2`