Problem 38 - Calculus: Early Transcendentals

Question

Determine the infinite limit. \( \displaystyle \lim_{x \to \pi^-}\cot x \)

Step-by-Step Solution

Answer

The limit is \( \infty \).

Step 1: Understanding the Cotangent Function

The cotangent function, \( \cot x \), is defined as \( \frac{1}{\tan x} \) or \( \frac{\cos x}{\sin x} \). As \( x \) approaches \( \pi \) from the left (i.e., \( \pi^- \)), we need to consider the behavior of \( \sin x \) and \( \cos x \).

Step 2: Behavior of Sine and Cosine Near \( \pi \)

As \( x \to \pi^- \), \( \sin x \to 0^- \) because the sine of values just less than \( \pi \) is slightly negative. Concurrently, \( \cos x \to -1 \) since cosine at \( \pi \) is \(-1\).

Step 3: Analyzing the Limit

The quotient \( \frac{\cos x}{\sin x} \) becomes \( \frac{-1}{0^-} \). Since \( \sin x \) is approaching \( 0 \) from the negative side, dividing by an increasingly small negative number results in a positively large value.

Step 4: Conclusion on Infinite Limit

Thus, the limit \( \lim_{x \to \pi^-}\cot x \) approaches \( \infty \) because the negative denominator causes the fraction to become positively large as it nears zero.

Key Concepts

Cotangent Function

The cotangent function, denoted as \( \cot x \), is a fundamental trigonometric function closely related to the tangent function. It is defined as the reciprocal of the tangent: \( \cot x = \frac{1}{\tan x} \). Alternatively, you can express it using sine and cosine functions as \( \cot x = \frac{\cos x}{\sin x} \).

This definition implies that the behavior of \( \cot x \) is heavily dependent on the values of \( \sin x \) and \( \cos x \). This is particularly crucial when approaching values where \( \sin x \) tends to zero, as the division by a very small number considerably amplifies the output of the function.

Understanding the cotangent function is key when analyzing limits involving \( \cot x \), especially when evaluating infinite limits as the input approaches critical angles like \( \pi \), where the sine function approaches zero.

Sine and Cosine Behavior

To fully grasp the behavior of trigonometrical functions around important angles like \( \pi \), we need to understand the characteristics of sine and cosine.

Sine and cosine functions are periodic with a period of \( 2\pi \). This means they repeat their values every \( 2\pi \) intervals. Specifically, sine reaches zero at integral multiples of \( \pi \) (e.g., \( 0, \pi, 2\pi, \ldots \)) and fluctuates between -1 and 1.

As \( x \to \pi^- \) (approaching \( \pi \) from the left):

• \( \sin x \) approaches 0 but stays slightly negative because it is crossing from positive to zero along the negative side.
• \( \cos x \) approaches \(-1\), maintaining its negative value since cosine is \(-1\) precisely at \( \pi \).

In this scenario, the minute negative value of \( \sin x \) significantly impacts the cotangent function, influencing its limit to become extremely large.

Trigonometric Limits

Trigonometric limits often involve understanding the behavior of trigonometric functions near critical values, such as those where the function becomes undefined or approaches infinity.

The limit \( \lim_{{x \to \pi^-}} \cot x \) is a classic example of an infinite limit. To solve it, we analyze the expression \( \frac{\cos x}{\sin x} \) as \( x \) nears \( \pi \) from the left. As discussed, \( \sin x \) approaches 0 from the negative side creating a scenario where dividing by this small value results in a large positive number.

Why does this happen? Because:

• The denominator, \( \sin x \), getting smaller and smaller (toward 0) means the division's overall value expands massively.
• Since \( \sin x \) is negative, dividing a negative \(-1\) (cosine's value here) by a negative small value results in a positive large value.

This behavior leads to the conclusion that \( \lim_{{x \to \pi^-}} \cot x = +\infty \), demonstrating the impact of trigonometric limits on infinite calculations.