Problem 33 - Thomas Calculus

Question

Find the limits in Exercises \(21-42\) $$\lim _{x \rightarrow 0} \frac{x-x \cos x}{\sin ^{2} 3 x}$$

Step-by-Step Solution

Answer

The limit is \( \frac{1}{12} \).

Step 1: Understand the Limit Expression

We are asked to evaluate \( \lim _{x \rightarrow 0} \frac{x-x \cos x}{\sin ^{2} 3 x} \). The expression involves trigonometric functions and an indeterminate form \( \frac{0}{0} \). This suggests we may need to use L'Hôpital's Rule or trigonometric identities to simplify.

Step 2: Apply Trigonometric Identities

Rewrite the expression \( x - x \cos x = x(1 - \cos x) \). Similarly, express \( \sin^2 3x \) as \( \left(\sin 3x\right)^2 \). The limit then becomes \( \lim_{x \to 0} \frac{x(1 - \cos x)}{\sin^2 3x} \).

Step 3: Use Trigonometric Limit and Identity

Recall the limit \( \lim_{x \to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2} \). We use this to simplify the expression further. Also, use the identity \( \sin 3x = 3x - \frac{(3x)^3}{6} + \cdots \approx 3x \) for small \( x \).

Step 4: Simplify Using L'Hôpital's Rule

Apply L'Hôpital's Rule to the expression \( \lim_{x \to 0} \frac{x(1 - \cos x)}{(3x)^2} \). Differentiate the numerator and denominator with respect to \( x \).

Step 5: Differentiate and Simplify Further

The numerator derivative is \( 1 - \cos x + x \sin x \). The denominator \( (3x)^2 = 9x^2 \) becomes \( 18x \). Evaluate the limit of this new expression \( \lim_{x \to 0} \frac{1 - \cos x + x \sin x}{18x} \).

Step 6: Evaluate the Limit

As \( x \to 0 \), \( 1 - \cos x \sim \frac{1}{2}x^2 \) based on the previous identity. The term \( x \sin x \) is \( x^2 \) approximately. Both terms in the numerator yield \( \frac{x^2}{2} + x^2 = \frac{3x^2}{2} \). This simplifies to \( \lim_{x \to 0} \frac{3x^2/2}{18x} = \frac{3}{36} = \frac{1}{12} \).

Step 7: Final Evaluation and Conclusion

The expressions have been carefully applied with trigonometric identities and derivatives. The limit resolves to a finite value.

Key Concepts

L'Hôpital's Rule

L'Hôpital's Rule is a powerful tool in calculus used to find limits, particularly when dealing with indeterminate forms like 0/0 or ∞/∞. When you encounter a limit that results in an indeterminate form, L'Hôpital's Rule allows you to differentiate the numerator and the denominator separately. Then, you can evaluate the limit again by substituting the derivatives back into the original expression. This rule simplifies the process, but it's important to remember:

• L'Hôpital's Rule can only be applied if initial substitution yields an indeterminate form.
• You must verify that the limits of the derivatives still lead to a meaningful result.
• Sometimes, you may need to apply the rule more than once if the resulting expression is still indeterminate.

In the given exercise, L'Hôpital's Rule was used to navigate the limit of an expression resulting in 0/0. Both the numerator and the denominator were differentiated, simplifying the limit evaluation to a rational result.

Trigonometric Limits

Trigonometric limits often appear in calculus when dealing with functions involving sine, cosine, and similar trigonometric identities. Remembering some key trigonometric limits can greatly simplify many calculus problems. For example, one very useful limit is:

• \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \).
• \( \lim_{x \to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2} \)

These help in approximating the behavior of trig functions near zero.
In the problem, we used the identity \( \sin 3x \approx 3x \) for small \( x \), helping us transition into a more straightforward expression to differentiate.
Knowing these limits and identities can make solving these types of problems more intuitive.

Indeterminate Forms

Indeterminate forms arise in calculus when it's not immediately clear what the limit of an expression might be. They're typically in forms like 0/0, ∞/∞, 0×∞, and others that don't lend themselves to simple algebraic manipulation.
When you encounter indeterminate forms, several strategies can be employed, such as:

• Simplifying the expression using algebraic techniques or trigonometric identities.
• Applying L'Hôpital's Rule to differentiate and re-evaluate the limit.
• Using series expansions or symmetrical properties of functions to resolve the ambiguity.

In the given exercise, both the numerator \(x - x\cos x\) and the denominator \(\sin^2 3x\) led to a 0/0 form as \(x\) approached zero. By recognizing this indeterminate form, we applied L'Hôpital's Rule and utilized trigonometric identities to accurately compute the limit.