Question

Determine the following limits. $$\lim _{x \rightarrow 0^{+}}(-10 \cot x)$$

Short answer

Answer: As x approaches 0 from the positive side, the function approaches negative infinity. \(\lim_{x \rightarrow 0^{+}}(-10 \cot x) = -\infty\)

Step-by-step solution

Examine the cotangent function

The cotangent function \(\cot x\) is the reciprocal of the tangent function, that is, $$\cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x}$$ Notice that \(\cot x\) is undefined when \(\sin x = 0\), which occurs at integer multiples of \(\pi\).

Rewrite the limit in terms of sine and cosine

Before proceeding with the limit, let's express the original function in terms of sine and cosine: $$\lim_{x \rightarrow 0^{+}}(-10 \cot x) = \lim_{x \rightarrow 0^{+}}\left(-10\frac{\cos x}{\sin x}\right)$$

Evaluate the one-sided limit

The one-sided limit means we are approaching \(x=0\) from the positive side, which means \(x\) is a very small positive number. Since \(\lim_{x \rightarrow 0^{+}} \cos x = 1\) and \(\lim_{x \rightarrow 0^{+}} \sin x = 0^{+}\), we have: $$\lim_{x \rightarrow 0^{+}}\left(-10\frac{\cos x}{\sin x}\right) = -10\frac{1}{0^{+}}$$

Determine the behavior of the limit

When dividing by a very small positive number, the expression becomes very large. Since the division is multiplied by -10, the final limit is: $$\lim_{x \rightarrow 0^{+}}(-10 \cot x) = -\infty$$ meaning that when \(x\) approaches 0 from the positive side, this function approaches negative infinity.

Key concepts

Cotangent Function

Understanding the cotangent function is crucial when dealing with trigonometric limits. Essentially, the cotangent is one of the six fundamental trigonometric functions and is denoted as \( \text{cot}(x) \). It represents the reciprocal of the tangent function, which implies that \( \text{cot}(x) = \frac{1}{\text{tan}(x)} = \frac{\text{cos}(x)}{\text{sin}(x)} \).

Since cotangent is defined as the ratio of the adjacent side to the opposite side in a right-angled triangle, it is undefined whenever the sine function equals zero, as division by zero is not allowed. The sine function equals zero at integer multiples of \( \text{pi} \), which are precisely the points where the cotangent function exhibits discontinuities or jumps from positive to negative infinity or vice versa. This behavior profoundly affects the limits as \( x \) approaches any of those points.

Limits of Trigonometric Functions

When dealing with limits of trigonometric functions, we're essentially trying to find the value that a trigonometric function approaches as the input gets infinitely close to a certain number. These limits can often be resolved by transforming the original function into a more manageable form using trigonometric identities.

For example, a limit involving the cotangent function might be transformed into a limit involving sine and cosine, as they are more fundamental and their behavior near certain critical points, like zero, is well understood. It's important to note that these transformations serve not only to simplify the expressions but also to help us avoid indeterminate forms that can complicate limit evaluation.

Indeterminate Forms

An indeterminate form is an expression that does not have a specific limit or value in its current form. In calculus, common indeterminate forms include \( \frac{0}{0} \), \( \frac{\text{infinity}}{\text{infinity}} \), 0 times infinity, \( \text{infinity} - \text{infinity} \), \( 0^0 \), \( \text{infinity}^0 \), and \( 1^\text{infinity} \).

These forms typically arise in the context of limits when the terms in a function approach values that make the function’s limit uncertain. To resolve these indeterminate forms and find the actual limit, we can apply L'Hôpital's rule, algebraic manipulation, or trigonometric identities. In the given exercise, we encounter the form \( \frac{1}{0^{+}} \), which suggests an infinite limit rather than a specific finite value.

Infinite Limits

Infinite limits occur when the value of a function grows without bound as the input approaches a particular point. It is to say, the function either goes to positive infinity (\( +\text{infinity} \)) or negative infinity (\( -\text{infinity} \)), depending on the direction of the growth.

In the given exercise, the negative sign in front of the cotangent function indicates that as the value of the cotangent grows larger, the entire expression moves in the negative direction, hence approaching negative infinity. Understanding infinite limits is essential, especially in one-sided limits where we're only concerned with the behavior of the function as it approaches from either the left or the right side, not both. This distinction is crucial in sketching accurate graphs of functions and in understanding their behavior near points where they are not defined.