Problem 19
Question
Find the limits. (If in doubt, look at the function's graph.) $$\lim _{x \rightarrow \infty} \csc ^{-1} x$$
Step-by-Step Solution
Verified Answer
The limit is 0.
1Step 1: Understanding the Function
The function given is the inverse cosecant, denoted as \( \csc^{-1} x \). The inverse cosecant is defined for \( x \leq -1 \) or \( x \geq 1 \). As \( x \to \infty \), we are interested in the behavior of \( \csc^{-1} x \).
2Step 2: Revisiting Cosecant
Recall that \( \csc x = \frac{1}{\sin x} \). The inverse \( \csc^{-1} x \) reverts this relationship, determining the angle \( \theta \) for which \( \csc \theta = x \). For very large values of \( x \), \( \theta \) approaches \( 0^+ \) because \( \csc \theta \) increases as \( \theta \) approaches \( 0 \) from the positive side.
3Step 3: Finding the Angle Limit
As \( x \to \infty \), \( \csc^{-1} x \) approaches the angle \( \theta \to 0^+ \). This is because the reciprocal of a very large number (\( x \)) is very small, meaning the sine of that angle approaches zero.
4Step 4: Stating the Limit
Thus, we deduce that \( \lim_{x \to \infty} \csc^{-1} x = 0 \). This limit represents the behavior of the inverse cosecant function as \( x \) becomes infinitely large.
Key Concepts
Inverse Trigonometric FunctionsCosecant FunctionBehavior at Infinity
Inverse Trigonometric Functions
When we talk about inverse trigonometric functions, we refer to functions that "undo" the trigonometric functions. They allow us to determine an angle given a specific trigonometric value. The common inverse trigonometric functions include \( ext{arcsin}, \ ext{arccos}, \ ext{arctan}, \ ext{arcsec}, \ ext{arccsc}, \ \text{and} \ ext{arccot} \). These functions are incredibly useful in calculus and have distinct properties and ranges.
For instance, the inverse cosecant function, \( ext{csc}^{-1} x \), is the inverse of the cosecant function. It is defined for \( x \leq -1 \) or \( x \geq 1 \). When using \( ext{csc}^{-1} \), we find the angle \( \theta \) such that \( \csc \theta = x \). This property is crucial when determining the limit behavior of \( \text{csc}^{-1} x \) as \( x \) becomes increasingly large.
Understanding these inverse functions allows us to explore deeper concepts in math, especially those related to the limits and behaviors of different functions.
For instance, the inverse cosecant function, \( ext{csc}^{-1} x \), is the inverse of the cosecant function. It is defined for \( x \leq -1 \) or \( x \geq 1 \). When using \( ext{csc}^{-1} \), we find the angle \( \theta \) such that \( \csc \theta = x \). This property is crucial when determining the limit behavior of \( \text{csc}^{-1} x \) as \( x \) becomes increasingly large.
Understanding these inverse functions allows us to explore deeper concepts in math, especially those related to the limits and behaviors of different functions.
Cosecant Function
The cosecant function, denoted \( \csc x \), is the reciprocal of the sine function. This means \( \csc x = \frac{1}{\sin x} \). The cosecant function is undefined for values where \( \sin x = 0 \), creating certain asymptotes in its graph.
In practical application, it shifts points from the sine graph into regions where the sine value becomes very small or zero, which results in very high or undefined values for the cosecant. As an example, for a small angle \( \theta \) close to zero, \( \csc \theta \) becomes very large since the sine of that angle is very small.
This behavior is crucial when considering limits at infinity for the inverse cosecant. The reciprocal nature of the function guides us to understand that as \( x \to \infty \), the approaching angle \( \theta \) where \( \csc \theta = x \) will be close to zero.
In practical application, it shifts points from the sine graph into regions where the sine value becomes very small or zero, which results in very high or undefined values for the cosecant. As an example, for a small angle \( \theta \) close to zero, \( \csc \theta \) becomes very large since the sine of that angle is very small.
This behavior is crucial when considering limits at infinity for the inverse cosecant. The reciprocal nature of the function guides us to understand that as \( x \to \infty \), the approaching angle \( \theta \) where \( \csc \theta = x \) will be close to zero.
Behavior at Infinity
In calculus, the behavior of functions as they approach infinity or zero is a key area of study. The limit \( \lim_{x \to \infty} f(x) \) helps us understand how a function behaves when \( x \) becomes very large. This concept is crucial when finding limits for inverse trigonometric functions like \( \csc^{-1} x \).
The function \( \csc^{-1} x \) specifically aims to find the angle \( \theta \) such that its cosecant equals \( x \). As \( x \to \infty \), the angle \( \theta \) becomes very close to zero. Here, it’s important to notice that the reciprocal of a large number becomes very small, indicating that the sine of this small angle also approaches zero.
The function \( \csc^{-1} x \) specifically aims to find the angle \( \theta \) such that its cosecant equals \( x \). As \( x \to \infty \), the angle \( \theta \) becomes very close to zero. Here, it’s important to notice that the reciprocal of a large number becomes very small, indicating that the sine of this small angle also approaches zero.
- \( \lim_{x \to \infty} \csc^{-1} x = 0 \): As \( x \) becomes very large, the output approaches 0.
- Understanding this behavior helps in predicting the limits and attributes of other similar inverse functions.
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