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(A) 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.$1$. T

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$b=2 \log |a|$

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(A) 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.$1$. The value of the function at $x=0$ is $f(0) = a^2 \cos^2(0) + b^2 \sin^2(0) = a^2(1) + b^2(0) = a^2$.$2$. The right-hand limit $(RHL)$ as $x \rightarrow 0^+$ is $\lim_{x \rightarrow 0^+} e^{ax+b} = e^{a(0)+b} = e^b$.$3$. The left-hand limit $(LHL)$ as $x \rightarrow 0^-$ is $\lim_{x \rightarrow 0^-} (a^2 \cos^2 x + b^2 \sin^2 x) = a^2(1) + b^2(0) = a^2$.For continuity,$LHL = RHL = f(0)$,so $e^b = a^2$.Taking the natural logarithm on both sides,we get $\ln(e^b) = \ln(a^2)$,which simplifies to $b = 2 \ln |a|$ or $b = 2 \log |a|$.

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