Determine the following limits.


$\underset{x\to 0^{-}}{\lim }\csc \left(x\right)$

Verified step by step guidance

1. Step 1: Understand the function involved. The cosecant function, \( \csc(x) \), is the reciprocal of the sine function, so \( \csc(x) = \frac{1}{\sin(x)} \).
2. Step 2: Analyze the behavior of \( \sin(x) \) as \( x \to 0^{-} \). As \( x \) approaches 0 from the left, \( \sin(x) \) approaches 0. However, since \( \sin(x) \) is negative just to the left of 0, \( \sin(x) \to 0^{-} \).
3. Step 3: Consider the reciprocal function \( \csc(x) = \frac{1}{\sin(x)} \). As \( \sin(x) \to 0^{-} \), the value of \( \frac{1}{\sin(x)} \) becomes very large in the negative direction.
4. Step 4: Conclude the behavior of \( \csc(x) \) as \( x \to 0^{-} \). Since \( \sin(x) \to 0^{-} \), \( \csc(x) \to -\infty \).
5. Step 5: Summarize the limit. Therefore, \( \lim_{x \to 0^{-}} \csc(x) = -\infty \).

Key Concepts

Limits

Limits are fundamental concepts in calculus that describe the behavior of a function as its input approaches a certain value. They help in understanding how functions behave near points of interest, including points where they may not be defined. In this case, we are interested in the limit of the cosecant function as x approaches 0 from the left.

Cosecant Function

The cosecant function, denoted as csc(x), is the reciprocal of the sine function, defined as csc(x) = 1/sin(x). It is important to note that csc(x) is undefined wherever sin(x) equals zero, which occurs at integer multiples of π. Understanding the behavior of the cosecant function near these points is crucial for evaluating limits involving csc.

One-Sided Limits

One-sided limits refer to the value that a function approaches as the input approaches a specific point from one side, either the left (denoted as x → a⁻) or the right (denoted as x → a⁺). In this question, we are specifically looking at the left-hand limit of csc(x) as x approaches 0, which requires analyzing the function's behavior as x gets closer to 0 from negative values.