Determine the following limits.
lim x→0^− 2 / tan x

Verified step by step guidance

1. Identify the limit expression: \( \lim_{{x \to 0^-}} \frac{2}{\tan x} \).
2. Recognize that as \( x \to 0^- \), \( \tan x \to 0^- \) because the tangent function approaches zero from the negative side.
3. Understand that \( \frac{2}{\tan x} \) will approach infinity or negative infinity depending on the sign of \( \tan x \) as \( x \to 0^- \).
4. Since \( \tan x \to 0^- \), the expression \( \frac{2}{\tan x} \) will approach negative infinity.
5. Conclude that the limit is \( -\infty \) as \( x \to 0^- \).

Key Concepts

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps 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 function as x approaches 0 from the left (denoted as x→0^−).

Tangent Function

The tangent function, tan(x), is a periodic function defined as the ratio of the sine and cosine functions: tan(x) = sin(x)/cos(x). It has specific properties, including vertical asymptotes where cos(x) = 0, which occur at odd multiples of π/2. Understanding the behavior of tan(x) near x = 0 is crucial for evaluating the limit in the question.

One-Sided Limits

One-sided limits refer to the value that a function approaches as the input approaches a specific point from one side only, either from the left (denoted as x→c^−) or from the right (denoted as x→c^+). In this problem, we are specifically looking at the left-hand limit as x approaches 0, which can yield different results than the right-hand limit, especially for functions with discontinuities.