Question

Determine the values of $a,b,c$ for which the function defined by
$f\left(x\right)=\frac{\sin (a+1)x+\sin x}{x},forx<0$
$c,forx=0$
$=\frac{{(x+bx)}^{1/2}-x^{1/2}}{{bx}^{1/2}},forx>0$
Is continuous at $x=0$

Answer 1

$\underset{x\to 0}{lim}\frac{\sin (a+1)a+\sin x}{x}$

$=\underset{x\to 0}{lim}\frac{\sin (a+1)x}{(a+1)x}\times a+1+\underset{x\to 0}{lim}\frac{\sin x}{x}$

$=a+1+1=a+2$

$R.H.L$

$\underset{x\to 0}{lim}\frac{\sqrt{x+bx}-\sqrt{x}}{b\sqrt{x}}$

$=\frac{\sqrt{1+b}-1}{b}$

for $f\left(x\right)$ to be continuous at $x=0$

$L.H.L=R.H.L=f\left(0\right)$

$\Rightarrow a+2=\frac{\sqrt{1+b}-1}{b}=C$

One more condition is required to final values of $a,b,c$