If f x -sin a-1 x -2sin x x -x0 is continuous at x-0- then find the values of a and b-

At x = 0 function is continuous, lim_x→ 0^+f(x)=lim_x→ 0^ f(x)=f(0) RHL: =lim_x→ 0^+f(x)=lim_x→ 0f(0+h)=lim_x→ 0√(1+bh) 1h×√(1+bh)+1√(1+bh)+1 ...

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Continuity of a Function

If f x =sin...

Question

If $f (x) = \frac{sin (a + 1) x + 2 sin x}{x}, x < 0 = 2, x = 0 = \frac{\sqrt{1 + b x} - 1}{x}, x > 0$
is continuous at $x = 0,$ then find the values of $a$ and $b .$

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f (x) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ \begin{matrix} \frac{sin (a + 1) x + 2 sin x}{x} & , & x < 0 2 & , & x = 0 \frac{\sqrt{1 + b x} - 1}{x} & , & x > 0 \end{matrix}

x = 0

a

b

f (x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} \frac{sin (a + 1) x}{x} & , i f x < 0 c & , i f x = 0 \frac{\sqrt{x + b x^{2}} - \sqrt{x}}{b x^{3 / 2}} & , i f x > 0 \end{matrix} ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

x = 0

a, b

c

f (x) ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} \frac{e^{x} - 1 + a x}{x^{2}}, x > 0 b x = 0 \frac{sin \frac{x}{2}}{x} x < 0 \end{matrix}

a + b

f (x)

x = 0

a, b

c

f (x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} \frac{sin (a + 1) x + sin x}{x} & , x < 0 c & , x = 0 \frac{\sqrt{x + b x^{2}} - \sqrt{x}}{x^{3 / 2}} & , x > 0 \end{matrix}

x = 0

f (x) = \{\begin{matrix} \frac{\sin (a + 1) x + \sin x}{x}, & x < 0 \\ c, & x = 0 \\ \frac{\sqrt{x + b x^{2}} - \sqrt{x}}{b x \sqrt{x}}, & x > 0 \end{matrix}

- \frac{3}{2}

\frac{1}{2}

- \frac{3}{2}

- \frac{1}{2}

- \frac{3}{2}

\frac{1}{2}

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