Problem 16 - Single Variable Calculus

Question

Find the limit. $$ \lim _{x \rightarrow 0^{+}} \cot 2 x $$

Step-by-Step Solution

Answer

The short answer is that the limit does not exist as it approaches \(\infty\). This is because the expression simplifies to \(\frac{1}{0}\) when evaluating the limit as \(x\) approaches 0 from the positive side.

Step 1: Rewrite the cotangent function.

Recall that the cotangent function is the reciprocal of the tangent function: \[\cot(x) = \frac{1}{\tan(x)}\] Rewrite the function inside the limit using this definition: \[\lim_{x \rightarrow 0^+} \cot(2x) = \lim_{x \rightarrow 0^+} \frac{1}{\tan(2x)}\]

Step 2: Apply trigonometric identity for tangent function.

We need to apply the double angle identity for the tangent function to simplify the expression: \[\tan(2x) = \frac{2\tan(x)}{1-\tan^2(x)}\] Now we rewrite the limit with the double angle identity applied: \[\lim_{x \rightarrow 0^+} \frac{1}{\tan(2x)} = \lim_{x \rightarrow 0^+} \frac{1}{\frac{2\tan(x)}{1-\tan^2(x)}}\]

Step 3: Invert the function and simplify the expression.

To simplify the expression, invert the function and cancel common factors: \[\lim_{x \rightarrow 0^+} \frac{1}{\frac{2\tan(x)}{1-\tan^2(x)}} = \lim_{x \rightarrow 0^+} \frac{1-\tan^2(x)}{2\tan(x)}\]

Step 4: Evaluate the limit.

Now that the expression is simplified, evaluate the limit as \(x\) approaches 0 from the positive side: \[\lim_{x \rightarrow 0^+} \frac{1-\tan^2(x)}{2\tan(x)} = \frac{1-\tan^2(0)}{2\tan(0)} = \frac{1-0}{2(0)}\] The expression becomes \(\frac{1}{0}\), which indicates that the limit is undefined. Therefore, the limit for the given function as \(x\) approaches 0 from the positive side does not exist.

Key Concepts

Trigonometric Identities

Trigonometric identities are mathematical equations that relate the angles and lengths in a right triangle. They help us simplify trigonometric expressions. Understanding these identities can be very useful in calculus, especially when you need to find limits of trigonometric functions.

• One common identity is the reciprocal relationship between the cotangent and tangent functions: \( \cot(x) = \frac{1}{\tan(x)} \). This means the cotangent is the inverse of tangent, turning division into multiplication.
• Another important identity is the tangent double angle formula: \( \tan(2x) = \frac{2\tan(x)}{1-\tan^2(x)} \). This formula simplifies expressions where the tangent function is applied to a doubled angle.

Understanding these identities allows us to transform complex trigonometric expressions into simpler ones. Each identity offers a stepping stone to evaluate the behaviors of functions in different limits.

Cotangent Function

The cotangent function, denoted as \( \cot(x) \), plays a special role in trigonometry and calculus. It is defined as the ratio of the adjacent to the opposite side of a right triangle or the reciprocal of the tangent function.

• In terms of sine and cosine, \( \cot(x) = \frac{\cos(x)}{\sin(x)} \).
• The cotangent function is undefined when the sine function equals zero, as this would mean division by zero.

The cotangent function, like other trigonometric functions, is periodic, repeating its values in regular intervals. It is especially important in calculus because it introduces limits that are often undefined, leading to intriguing challenges and deeper understanding.

Undefined Limit

An undefined limit occurs when we cannot determine a finite result as a limit of a function. This situation often arises when evaluating limits results in division by zero.

• Consider the limit \( \lim_{x \rightarrow 0^+} \cot(2x) \). As you approach this limit, the function tends towards \( \frac{1}{0} \), indicating it's undefined.
• When evaluating limits approaching zero, functions like tangent and cotangent are particularly troublesome because they oscillate rapidly.

Recognizing an undefined limit is crucial in calculus, as it signals the presence of an asymptote or a special behavior where normal rules may need adapting. Learning to identify undefined limits can help in understanding the behavior of a function around particular points.