Consider the graph of y=cot^−1 x(see Section 1.4) and determine the following limits using the graph.
lim x→−∞ cot^−1x

검증된 단계별 안내

Step 1: Understand the function \( y = \cot^{-1}(x) \). The inverse cotangent function, \( \cot^{-1}(x) \), is the angle whose cotangent is \( x \). It is defined for all real numbers and its range is \((0, \pi)\).

Step 2: Consider the behavior of \( \cot^{-1}(x) \) as \( x \to -\infty \). As \( x \) becomes very large in the negative direction, the angle whose cotangent is \( x \) approaches \( \pi \).

Step 3: Visualize the graph of \( y = \cot^{-1}(x) \). As \( x \to -\infty \), the graph approaches the horizontal asymptote at \( y = \pi \).

Step 4: Use the graph to determine the limit. The graph shows that as \( x \to -\infty \), \( \cot^{-1}(x) \) approaches \( \pi \).

Step 5: Conclude that \( \lim_{x \to -\infty} \cot^{-1}(x) = \pi \).

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주요 개념

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Inverse Cotangent Function

The inverse cotangent function, denoted as cot^−1(x), is the function that returns the angle whose cotangent is x. It is defined for all real numbers and has a range of (0, π). Understanding this function is crucial for analyzing its behavior as x approaches different limits.

Limits in Calculus

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. Evaluating limits helps in understanding the behavior of functions at boundaries, including infinity. In this case, we are interested in the limit of cot^−1(x) as x approaches negative infinity.

Behavior of Functions at Infinity

The behavior of functions as they approach infinity or negative infinity is essential for understanding their long-term trends. For the cotangent function, as x approaches negative infinity, the value of cot^−1(x) approaches π. This concept helps in predicting the output of the function without direct computation.

Consider the graph of y=cot^−1 x(see Section 1.4) and determine the following limits using the graph.lim x→−∞ cot^−1x