Problem 21 - Calculus of a Single Variable

Question

Finding a Limit In Exercises \(7-26\) , find the limit (if it exists). If it does not exist, explain why. $$ \lim _{x \rightarrow \pi} \cot x $$

Step-by-Step Solution

Answer

The limit as x approaches \(\pi\) for \(\cot x\) does not exist.

Step 1: Understand the function

The cotangent function is undefined at \(x = \pi\). This is because \(\cot x = \frac{\cos x}{\sin x}\), and \(\sin \pi = 0\), hence leading to a division by zero, which is undefined.

Step 2: Evaluate the limit from left and right

The limit of the cotangent function at a point where it's undefined doesn't necessarily mean the limit itself is undefined. As x approaches \(\pi\) from both the left side (denoted as \(\lim _{x \rightarrow \pi^{-}} \cot x\)) and the right side (denoted as \(\lim _{x \rightarrow \pi^{+}} \cot x\)), it is observed that the cotangent values approach negative and positive infinity respectively. Therefore, the two-sided limit does not exist.

Step 3: Conclusion

If the limits from both sides of the value aren't the same, then the limit of the function at that point does not exist.

Key Concepts

Cotangent Function

The cotangent function, usually written as \text{cot(x)}, is one of the six primary trigonometric functions. It is defined as the ratio of the adjacent side to the opposite side in a right-angled triangle, or, more commonly in calculus, as the ratio of the cosine function to the sine function: \[ \text{cot}(x) = \frac{\cos(x)}{\sin(x)} \]

It's important to understand that the cotangent function has an intimate relationship with the behavior of the sine and cosine functions. Since the cotangent function involves a division by \text{sin(x)}, whenever \text{sin(x)} equals zero, cotangent becomes undefined, creating what are called 'discontinuities' or 'singularities' at those points. In the unit circle, this occurs at integer multiples of \(\pi\). Between these discontinuities, the cotangent function is continuous and smoothly transitions from positive to negative infinity.

Two-Sided Limits

In calculus, two-sided limits are used to describe the behavior of a function as the input approaches a specific value from both the left (negative direction) and the right (positive direction). This concept is visually represented on a graph where the function near the point of interest is scrutinized from both sides. The limit exists if both one-sided limits―denoted as \(\lim_{x \rightarrow c^-}f(x)\) and \(\lim_{x \rightarrow c^+}f(x)\)―are equal and finite.

When working with the cotangent or any other trigonometric function, evaluating two-sided limits can be trickier due to their periodic nature and discontinuities. As we approach these critical points from both sides, if the behavior of the function markedly differs, this implies that a two-sided limit does not exist.

Undefined Limit

An undefined limit refers to a situation where a limit does not lead to a specific numerical value. This can happen due to various reasons such as the function approaching infinity, a difference in behavior from the left and right sides, or the function exhibiting oscillatory behavior without settling on a single value as the input approaches the desired point.

In the context of the cotangent function, an undefined limit often arises at points where \text{sin(x)} is zero, leading to a division by zero situation. For instance, at \(x = \pi\), \text{sin(x)} is zero and therefore \(\lim_{x \rightarrow \pi} \cot(x)\) is undefined because the outputs shoot off to infinity from one side and negative infinity from the other without aligning on a common value.