Problem 28 - Thomas Calculus

Question

Find the limits in Exercises \(21-42\) $$\lim _{x \rightarrow 0} 6 x^{2}(\cot x)(\csc 2 x)$$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Identify the Expression and Functions Involved

The expression we need to evaluate as the limit approaches zero is \(6x^2 \cot(x) \csc(2x)\). This expression involves trigonometric functions \(\cot(x)\), which is \(\frac{\cos(x)}{\sin(x)}\), and \(\csc(2x)\), which is \(\frac{1}{\sin(2x)}\).

Step 2: Substitute Trigonometric Functions with Their Identities

Rewrite the expression using trigonometric identities: \[6x^2 \cdot \frac{\cos(x)}{\sin(x)} \cdot \frac{1}{\sin(2x)}\] The identity for \(\sin(2x)\) is \(2\sin(x)\cos(x)\), so replace it within the expression:

Step 3: Simplify the Expression

Substituting \(\sin(2x) = 2\sin(x)\cos(x)\), the expression becomes: \[6x^2 \cdot \frac{\cos(x)}{\sin(x)} \cdot \frac{1}{2\sin(x)\cos(x)}\]This simplifies further to: \[\frac{6x^2}{2\sin^2(x)}\]Which reduces to: \[3x^2 \cot^2(x)\]

Step 4: Apply Limit to Simplified Expression

Now, evaluate the limit of the simplified expression:\[\lim_{x \to 0} 3x^2 \cot^2(x)\]Since \(\cot(x) = \frac{\cos(x)}{\sin(x)}\), apply \(\lim_{x \to 0} \frac{\cos(x)}{\sin(x)} = \infty\) because \(\sin(x)\) approaches zero as \(x\) approaches zero. So, \(x^2\) and \(x^2 \cot^2(x)\) are both further analyzed.

Step 5: Evaluate Limit Analysis with L'Hôpital's Rule

Transform the expression further and apply L'Hôpital's Rule if necessary. The limit transforms to:\[\lim_{x \to 0} 3x^2 \left(\frac{\cos(x)}{x}\right)^2\]Recognizing the expression from known limits as the forms \(\frac{1}{x^2}\) with \(x^2\) being negligible leads us to:\[\lim_{x \to 0} 3 \left(\frac{1}{x}\right)\left(\frac{\cos(x)}{\sin(x)}\right)^2 \ \rightarrow 0\]This approach incorporates the fact \(x\cot(x) \rightarrow 1\) as \(x\) approaches zero.

Key Concepts

L'Hôpital's Rule

L'Hôpital's Rule is a helpful tool in calculus for finding limits, especially when you encounter indeterminate forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). When direct substitution in a limit leads to such forms, L'Hôpital's Rule allows us to differentiate the numerator and the denominator separately. This differentiation process can make the limit simpler to evaluate.
To apply L'Hôpital's Rule, follow these steps:

• Ensure that the limit results in an indeterminate form such as \( \frac{0}{0} \).
• Differentiating the numerator and the denominator separately.
• Find the limit of this new expression.
• If the new expression is still indeterminate, repeat the process.

In the context of our problem, after simplifying the expression \( 3x^2 \cot^2(x) \), recognizing how \( x^2 \) behaves as \( x \) approaches zero is crucial. If you face challenges within this transformation involving trigonometric functions, L'Hôpital's Rule may assist in resolving lingering indeterminate forms.

Trigonometric Limits

Trigonometric limits often arise when expressions involve sine, cosine, tangent, or their reciprocal functions. Knowing some standard limits of these functions can simplify problems significantly.
Consider the limit \( \lim_{x \to 0} \frac{\sin(x)}{x} = 1 \). This is a fundamental limit often used in problems involving trigonometric functions as \( x \) approaches zero.
The original exercise involves the limit \( \lim _{x \rightarrow 0} 6 x^{2} (\cot x)(\csc 2 x) \). With trigonometric identities, you can rewrite this using known behaviors like \( \cot(x) = \frac{\cos(x)}{\sin(x)} \) and \( \csc(x) = \frac{1}{\sin(x)} \).

• Substitute using identities to transform the expression.
• Simplify to an expression where known limits can be applied easily.
• Analyze the behavior of sin and cos components as \( x \) approaches zero.

Understanding these trigonometric limits can guide you to the conclusion of how the function behaves, making the process less daunting.

Simplification of Expressions

Simplification of expressions makes complex calculus problems more manageable. In this process, the goal is to rewrite the original function using algebraic identities and simplifications to a form that is easier to handle.
In our example, the initial expression \( 6x^2 \cot(x) \csc(2x) \) transformed step by step:

• Rewrite trigonometric functions using their definitions: \( \cot(x) = \frac{\cos(x)}{\sin(x)} \) and \( \csc(2x) = \frac{1}{\sin(2x)} \).
• Simplification involves substituting \( \sin(2x) = 2\sin(x)\cos(x) \). This reduces the expression significantly by canceling out factors.
• Exploit these identities to combine and reduce the expression into simpler terms: \( \frac{6x^2}{2\sin^2(x)} = 3x^2 \cot^2(x) \).

Simplifying before applying limits makes it easier to identify indeterminate forms and apply calculus principles effectively. By breaking down the trigonometric aspects, one can reach the limit analysis stage with greater clarity.