Problem 27 - Thomas Calculus

Question

Find the limits in Exercises \(21-42\) $$\lim _{x \rightarrow 0} \frac{x \csc 2 x}{\cos 5 x}$$

Step-by-Step Solution

Answer

The limit is \( \frac{1}{2} \).

Step 1: Rewrite Trigonometric Expressions

First, rewrite the given limit expression using the identities for cosecant and cosine. The expression is \( \lim_{x \to 0} \frac{x \csc 2x}{\cos 5x} \). It can be written as \( \lim_{x \to 0} \frac{x}{\sin 2x} \cdot \frac{1}{\cos 5x} \).

Step 2: Apply Limit Properties

Use the limit property \( \lim_{x \to 0} \frac{\sin kx}{kx} = 1 \) for small angles. Rewrite \( \frac{x}{\sin 2x} \) in terms of this property: \( \lim_{x \to 0} \frac{x}{\sin 2x} = \lim_{x \to 0} \frac{1}{2} \cdot \frac{2x}{\sin 2x} = \frac{1}{2} \cdot 1 = \frac{1}{2} \).

Step 3: Evaluate the Remaining Limit

Now evaluate the limit \( \lim_{x \to 0} \frac{1}{\cos 5x} \). As \( x \to 0 \), \( \cos 5x \to \cos 0 = 1 \), so \( \lim_{x \to 0} \frac{1}{\cos 5x} = 1 \).

Step 4: Multiply the Results of Limits

Combine the results of the limits evaluated: \( \frac{1}{2} \times 1 = \frac{1}{2} \). Hence, the limit is \( \frac{1}{2} \).

Key Concepts

Trigonometric Limits

Trigonometric limits are crucial in calculus because they help us understand behavior near specific points, particularly zero. Often, we deal with expressions involving functions like sine, cosine, tangent, and their reciprocals. For expressions such as \( \lim_{x \to 0} \frac{\sin kx}{kx} = 1 \) and \( \lim_{x \to 0} \cos kx = 1 \), recognizing these fundamental limits simplifies evaluating complex expressions, especially those involving multiple trigonometric functions.To understand this, remember:

• Sine and tangent can be replaced using known identities \( \sin x \approx x \) for small \( x \).
• Cosine behaves more straightforwardly, moving towards 1 as \( x \) approaches zero.

Breaking down such expressions using identities and these limit properties enables easier computation. This clarity paves the way toward mastering challenging limit evaluations.

L'Hôpital's Rule

L'Hôpital's Rule is a powerful tool in calculus for resolving indeterminate forms like \( \frac{0}{0} \) and \( \frac{\infty}{\infty} \). This rule states that if you can rewrite your limit problem in either of these forms, you can take the derivatives of the numerator and the denominator separately, then reapply the limit.For example, if you encounter a problem like \( \lim_{x \to a} \frac{f(x)}{g(x)} \) where both \( f(x) \) and \( g(x) \) approach 0 as \( x \to a \), you can resolve this by checking the new limit formed: \[ \lim_{x \to a} \frac{f'(x)}{g'(x)}. \]Keep applying the derivative process until you achieve a solvable limit. This rule drastically simplifies the solutions of many complex trigonometric limits where directly evaluating might initially seem troublesome.

Small Angle Approximations

Small angle approximations help make complex trigonometric evaluations manageable, especially in limits. When dealing with very small angles, trigonometric functions like sine and cosine have simple linear approximations. These approximations are:

• \( \sin x \approx x \)
• \( \cos x \approx 1 - \frac{x^2}{2} \) (more accurate).

For calculations, replacing these functions with their approximations can simplify expressions drastically.In the given problem, the approximation \( \sin 2x \approx 2x \) for small \( x \) helps find the limit of expressions like \( \frac{x}{\sin 2x} \). This substitution gives you a pivotal insight leading to correct limit evaluation.Make sure to use these approximations keeping in mind their valid range (typically where \( x \) is close to zero), as this ensures computational accuracy and efficiency.