If the function f as defined below is continuous at x=0find the values of a,b and c `f(x) = {(sin (a+1)x +sin x)/x, x<0 and c, x=0, and (sqrt(x+bx^2)-sqrt(x))/(bx^(3/2)), x>0`

Options

A. `a=(-3)/(2), c=(1)/(2), b=0`
B. `a=(3)/(2), c=(1)/(2), b ne 0`
C. `a=-(3)/(2), c=(1)/(2) , b ne 0`
D. None of the above

Answer

C

Explanation

In th definition of the function, if b = 0, then f(x) will be underfined in `x gt 0.`
`because ` f(x) is continous at x = 0,
`therefore LHL=RHL=f(0)`
`rArr lim_(x to 0)(sin(a+1)x+sinx)/(x)=lim_(x to 0)(sqrt(x+bx^(2))-sqrt(x))/(bx^(3//2))=c`
`rArr lim_(x to 0)((sin(a+1)x)/(x)+(sinx)/(x))=lim_(x to0)(sqrt(1+bx)-1)/(bx)=c`
`rArr (a+1)+1=lim_(x to0)((1+bx)-1)/(bx (sqrt(1+bx)+1))=c`
`" "[ because lim_(x to0)(sin(a+1)x)/(x)xx((a+1))/((a+1))=a+1]`
`rArr a+2=lim_(x to 0)(1)/(sqrt(1+bx)+1)=c rArr a+2=(1)/(2)=c`
`therefore a=-(3)/(2), c=(1)/(2), b ne 0`