Determine the following limits.
lim x→−∞ e^x sin x

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Recognize that the problem involves finding the limit of the product of two functions as \( x \) approaches \(-\infty\).

Identify the two functions involved: \( e^x \) and \( \sin x \).

Recall that \( e^x \) approaches 0 as \( x \) approaches \(-\infty\).

Note that \( \sin x \) oscillates between -1 and 1 for all real \( x \).

Conclude that the product \( e^x \sin x \) approaches 0 as \( x \) approaches \(-\infty\) because \( e^x \) dominates the behavior of the product by approaching 0.

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