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

검증된 단계별 안내

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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Limits at Infinity

Limits at infinity involve evaluating the behavior of a function as the input approaches positive or negative infinity. Understanding how functions behave in these scenarios is crucial for determining their limits, especially when dealing with exponential and trigonometric functions.

Exponential Functions

Exponential functions, such as e^x, grow rapidly as x increases and approach zero as x decreases towards negative infinity. This characteristic is essential for analyzing the limit of e^x as x approaches negative infinity, which significantly influences the overall limit of the expression.

Trigonometric Functions

Trigonometric functions like sin(x) oscillate between -1 and 1, regardless of the value of x. This periodic behavior means that while sin(x) does not converge to a single value, its bounded nature plays a critical role in determining the limit of the product e^x sin(x) as x approaches negative infinity.

Determine the following limits.lim x→−∞ ex sin x