Let f(x) = begin{cases} (1 + |sin x|)^{a/|sin | vedclass

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(B) For $f(x)$ to be continuous at $x = 0$,we must have $\lim_{x 	o 0^-} f(x) = f(0) = \lim_{x 	o 0^+} f(x)$.First,calculate the left-hand limit $(L

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$2/3, e^{2/3}$

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(B) For $f(x)$ to be continuous at $x = 0$,we must have $\lim_{x 	o 0^-} f(x) = f(0) = \lim_{x 	o 0^+} f(x)$.First,calculate the left-hand limit $(LHL)$:$\lim_{x 	o 0^-} (1 + |\sin x|)^{a/|\sin x|} = e^{\lim_{x 	o 0^-} |\sin x| \cdot \frac{a}{|\sin x|}} = e^a$.Next,calculate the right-hand limit $(RHL)$:$\lim_{x 	o 0^+} e^{	an 2x/	an 3x} = e^{\lim_{x 	o 0^+} \frac{	an 2x}{	an 3x}} = e^{\lim_{x 	o 0^+} \frac{(	an 2x / 2x) \cdot 2x}{(	an 3x / 3x) \cdot 3x}} = e^{2/3}$.Since $f(0) = b$,we equate the limits:$e^a = b = e^{2/3}$.Thus,$a = 2/3$ and $b = e^{2/3}$.

Let $f(x) = \begin{cases} (1 + |\sin x|)^{a/|\sin x|}, & -\pi/6 < x < 0 \\ b, & x = 0 \\ e^{\tan 2x/\tan 3x}, & 0 < x < \pi/6 \end{cases}$. If $f$ is continuous at $x = 0$,then the values of $a$ and $b$ are respectively:

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