Determine the following limits.

lim x→−∞ (e^x cos x +3)

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Identify the behavior of each component of the function as \( x \to -\infty \).

Recognize that \( e^x \to 0 \) as \( x \to -\infty \) because the exponential function decays to zero.

Note that \( \cos x \) oscillates between -1 and 1 for all real numbers, including as \( x \to -\infty \).

Consider the product \( e^x \cos x \). Since \( e^x \to 0 \) and \( \cos x \) is bounded, \( e^x \cos x \to 0 \) as \( x \to -\infty \).

Combine the results: \( \lim_{x \to -\infty} (e^x \cos x + 3) = 0 + 3 = 3 \).

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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 end behavior, which can often simplify complex expressions.

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 limits involving exponential terms, particularly when combined with oscillatory functions like cosine.

Trigonometric Functions

Trigonometric functions, like cos x, oscillate between -1 and 1 regardless of the value of x. When evaluating limits that include trigonometric functions, it's important to recognize their bounded nature, which can influence the overall limit when combined with other terms.

Determine the following limits. lim x→−∞ (e^x cos x +3)