If f(x) = begin{cases} frac{sin(a+2)x + sin x | vedclass

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(D) For the function $f(x)$ to be continuous at $x = 0$, the left-hand limit, right-hand limit, and the value of the function at $x = 0$ must be equal

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(D) For the function $f(x)$ to be continuous at $x = 0$, the left-hand limit, right-hand limit, and the value of the function at $x = 0$ must be equal.$1$. Left-hand limit $(LHL)$: $\lim_{x 	o 0^-} f(x) = \lim_{x 	o 0} \left( \frac{\sin(a+2)x}{x} + \frac{\sin x}{x} ight) = (a+2) + 1 = a+3$.$2$. Right-hand limit $(RHL)$: $\lim_{x 	o 0^+} f(x) = \lim_{x 	o 0} \frac{(x+3x^2)^{1/3} - x^{1/3}}{x^{4/3}} = \lim_{x 	o 0} \frac{x^{1/3}((1+3x)^{1/3} - 1)}{x^{4/3}} = \lim_{x 	o 0} \frac{(1+3x)^{1/3} - 1}{x}$.Using the binomial expansion $(1+u)^n \approx 1 + nu$ for small $u$:$\lim_{x 	o 0} \frac{(1 + \frac{1}{3}(3x)) - 1}{x} = \lim_{x 	o 0} \frac{1+x-1}{x} = 1$.$3$. Value at $x=0$: $f(0) = b$.For continuity, $a+3 = b = 1$.Thus, $a = -2$ and $b = 1$.Therefore, $a+2b = -2 + 2(1) = 0$.

If $f(x) = \begin{cases} \frac{\sin(a+2)x + \sin x}{x} & ; x < 0 \\ b & ; x = 0 \\ \frac{(x+3x^2)^{1/3} - x^{1/3}}{x^{4/3}} & ; x > 0 \end{cases}$ is continuous at $x = 0$, then $a+2b$ is equal to

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