Use the graph of $\mathrm{g}(x)=\sin (4 x) e^{0.17 x}-2$ and its asymptote to find the following limits. If necessary, choose the most informative answer from $\infty,-\infty$, and "Does Not Exist." (a) $\lim _{x \rightarrow-\infty} g(x)=$ (b) $\lim _{x \rightarrow \infty} g(x)=$

Step 1 ：First, we need to understand the function $g(x)=\sin(4x)e^{0.17x}-2$. This function is a combination of a sine function and an exponential function.

Step 2 ：The sine function, $\sin(4x)$, oscillates between -1 and 1 as $x$ goes to $\pm\infty$. The frequency of oscillation is determined by the coefficient of $x$, which is 4 in this case.

Step 3 ：The exponential function, $e^{0.17x}$, approaches 0 as $x$ goes to $-\infty$ and approaches $\infty$ as $x$ goes to $\infty$.

Step 4 ：The constant term -2 shifts the entire function down by 2 units.

Step 5 ：Therefore, as $x$ goes to $-\infty$, the sine function oscillates between -1 and 1, but the exponential function approaches 0. So, the product of these two functions also approaches 0. Subtracting 2 from this, we get $-2$ as $x$ goes to $-\infty$.

Step 6 ：As $x$ goes to $\infty$, the sine function still oscillates between -1 and 1, but the exponential function approaches $\infty$. So, the product of these two functions oscillates between $-\infty$ and $\infty$. Subtracting 2 from this, we get that the function oscillates between $-\infty$ and $\infty$ as $x$ goes to $\infty$.

Step 7 ：Therefore, the limit of $g(x)$ as $x$ goes to $-\infty$ is $-2$, and the limit of $g(x)$ as $x$ goes to $\infty$ does not exist.

Step 8 ：So, the answers are: (a) $\lim_{x\to -\infty} g(x) = \boxed{-2}$, (b) $\lim_{x\to \infty} g(x) = \text{Does Not Exist}$.