If f(x) defined as given below is continuous | vedclass

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(C) Given that $f(x)$ is continuous on $R$,it must be continuous at all points,including $x=0$ and $x=3$. For continuity at $x=0$: $f(0) = \lim_{x 	o

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-$2$

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(C) Given that $f(x)$ is continuous on $R$,it must be continuous at all points,including $x=0$ and $x=3$. For continuity at $x=0$: $f(0) = \lim_{x 	o 0^+} f(x) = \lim_{x 	o 0^-} f(x)$. $f(0) = \sin(0) = 0$. $\lim_{x 	o 0^+} (x^2+a) = 0^2+a = a$. $\lim_{x 	o 0^-} \sin x = 0$. Thus,$a = 0$. For continuity at $x=3$: $f(3) = \lim_{x 	o 3^+} f(x) = \lim_{x 	o 3^-} f(x)$. $f(3) = b(3)+3 = 3b+3$. $\lim_{x 	o 3^+} (-3) = -3$. $\lim_{x 	o 3^-} (bx+3) = 3b+3$. So,$3b+3 = -3 \implies 3b = -6 \implies b = -2$. Therefore,$a+b = 0 + (-2) = -2$.

If $f(x)$ defined as given below is continuous on $R$,then the value of $a+b$ is equal to: $f(x) = \begin{cases} \sin x, & x \leq 0 \\ x^2+a, & 0 < x < 1 \\ b x+3, & 1 \leq x \leq 3 \\ -3, & x > 3 \end{cases}$

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