Question

Find the limit. \(\lim _{x \rightarrow 0^{-}} \operatorname{coth} x\)

Short answer

The limit of the function as x approaches 0 from the negative side is -1.

Step-by-step solution

Define the function

First, define the function given \[ \operatorname{coth} x = \frac{\operatorname{cosh} x}{\operatorname{sinh} x} \] where cosh is the hyperbolic cosine and sinh is the hyperbolic sine. Then it can be expressed as \[ \operatorname{coth} x = \frac{e^x + e^{-x}}{e^x - e^{-x}} \]

Apply limit using properties

Now, we need to find the limit as x approaches 0 from the negative side, which is written mathematically as \[ \lim _{x \rightarrow 0^{-}} \operatorname{coth} x \]. Substitute the expression for coth x from Step 1 to get \[ \lim _{x \rightarrow 0^{-}} \frac{e^x + e^{-x}}{e^x - e^{-x}} \]

Evaluate the limit

As x approaches 0 from the negative side, \(e^x\) approaches 1 whereas \(e^{-x}\) becomes a large positive number. So, the expression \[ \frac{e^x + e^{-x}}{e^x - e^{-x}} \] approaches \[ \frac{1 + \infty}{1 - \infty} = -1 \]

Key concepts

Hyperbolic Functions

Hyperbolic functions are mathematical functions that, similar to trigonometric functions, describe relationships between angles and sides of a hyperbola. The most commonly used hyperbolic functions are hyperbolic sine (\text{sinh}) and hyperbolic cosine (\text{cosh}), which are defined using exponential functions:

• \text{sinh}(x) = \frac{e^x - e^{-x}}{2}
• \text{cosh}(x) = \frac{e^x + e^{-x}}{2}

From these definitions, we can construct other hyperbolic functions such as hyperbolic cotangent (\text{coth}), which is the reciprocal of \text{tanh} (hyperbolic tangent):
\[ \text{coth}(x) = \frac{\text{cosh}(x)}{\text{sinh}(x)} = \frac{e^x + e^{-x}}{e^x - e^{-x}} \]
Understanding how these functions behave is key to solving problems involving them. For instance, \text{coth}(x) tends to become large and positive as x approaches zero from the positive side, and large and negative as x approaches zero from the negative side.

Limit Properties

Calculus is full of procedures that help us study the behavior of functions as they approach a certain value or infinity. One such procedure is the limit. When finding the limit of a function, we're concerned with the value that a function approaches, rather than its exact value at that point. This is especially useful for points where the function might not be well-defined.
Some essential properties of limits that help solve limit problems include the limit laws, which allow us to break down complex expressions into simpler parts. For example, we can find the limit of a sum, difference, product, or quotient of functions individually, as long as the limits exist.
\[ \text{For instance, if} \ \text{lim}_{x \rightarrow a} f(x) \text{ and } \text{lim}_{x \rightarrow a} g(x) \text{ both exist, then } \ \text{lim}_{x \rightarrow a} [f(x) + g(x)] = \text{lim}_{x \rightarrow a} f(x) + \text{lim}_{x \rightarrow a} g(x) \]
It's imperative to understand these properties to correctly apply them and find limits, whether the functions are algebraic, trigonometric, or hyperbolic in nature.

Approaching from the Negative Side

In calculus, when we say we're approaching a point 'from the negative side' or 'from the left', we're talking about taking a limit where the variable approaches the point of interest from values less than that point. For positive 'from the right', values greater than the point are taken into consideration.
This distinction is crucial because functions can behave differently from each side of a point. For instance, the hyperbolic cotangent function, \text{coth}(x), behaves in opposite manners as x approaches zero from the positive (\text{right}) and negative (\text{left}) sides. Symbolically, this is represented as x tends towards 0\textsuperscript{-} or 0\textsuperscript{+}.
Thus, when evaluating \[\text{lim}_{x \rightarrow 0^{-}} \text{coth}(x)\], one must consider the behavior of \text{coth} as x approaches zero, but specifically from the left. It requires an intuitive understanding of how the exponential components within the \text{coth} function (\text{e}^x and \text{e}^{-x}) will behave as x shifts closer to zero from negative values. Essentially, as x approaches zero from the negative side, \text{e}^x approaches 1, while \text{e}^{-x} grows towards infinity, resulting in the limit approaching -1.