Lim x→0 (1 - cos(x))/x^2 = 1/2

Definition

The expression $$ ext{lim}_{x o 0} \frac{1 - \cos(x)}{x^2} = \frac{1}{2}$$ represents a specific limit that showcases how the function behaves as x approaches 0. This limit is crucial because it illustrates the relationship between trigonometric functions and their approximations, especially in calculus, where understanding the behavior of functions near critical points is essential. Mastering this limit helps in various applications, including L'Hôpital's Rule and Taylor series expansions, as it serves as an example of how limits can simplify complex expressions.

5 Must Know Facts For Your Next Test

1. The limit $$\text{lim}_{x \to 0} \frac{1 - \cos(x)}{x^2}$$ is a classic example often used to illustrate the application of L'Hôpital's Rule or Taylor Series.
2. This limit evaluates to $$\frac{1}{2}$$, demonstrating that as x gets very close to 0, the ratio of the expression approaches this specific value.
3. The expression $$1 - \cos(x)$$ can be approximated using the Taylor series expansion for cosine, revealing its quadratic behavior near x = 0.
4. Understanding this limit is essential when dealing with more complex limits that involve trigonometric functions in calculus problems.
5. The result $$\frac{1}{2}$$ can be interpreted geometrically by considering the unit circle and the relationships between the angles and the lengths of segments.

Review Questions

• How can L'Hôpital's Rule be applied to evaluate the limit $$\text{lim}_{x \to 0} \frac{1 - \cos(x)}{x^2}$$?
  • To apply L'Hôpital's Rule to the limit $$\text{lim}_{x \to 0} \frac{1 - \cos(x)}{x^2}$$, we first confirm that both the numerator and denominator approach 0 as x approaches 0, resulting in an indeterminate form. We then differentiate the numerator, which becomes $$\sin(x)$$, and the denominator, which becomes $$2x$$. Re-evaluating the limit gives us $$\text{lim}_{x \to 0} \frac{\sin(x)}{2x}$$. This limit simplifies to $$\frac{1}{2}$$ since we know that $$\text{lim}_{x \to 0} \frac{\sin(x)}{x} = 1$$.
• Discuss how the Taylor Series expansion for cos(x) relates to evaluating the limit $$\text{lim}_{x \to 0} \frac{1 - \cos(x)}{x^2}$$.
  • The Taylor Series expansion for cos(x) around x = 0 is given by $$\cos(x) = 1 - \frac{x^2}{2} + O(x^4)$$. By substituting this expansion into our limit expression, we can rewrite it as $$\frac{1 - (1 - \frac{x^2}{2} + O(x^4))}{x^2} = \frac{x^2/2 - O(x^4)}{x^2}$$. As x approaches 0, the higher-order terms (like O(x^4)) become negligible compared to the leading term, leading to a simplified limit of $$\frac{1}{2}$$.
• Evaluate and explain how understanding this limit influences higher-level calculus concepts such as integration or differential equations.
  • Understanding the limit $$\text{lim}_{x \to 0} \frac{1 - \cos(x)}{x^2} = \frac{1}{2}$$ is foundational for grasping more advanced concepts in calculus. It serves as a stepping stone for applying L'Hôpital's Rule and exploring convergence in series. Moreover, knowing how trigonometric functions behave near critical points aids in solving integrals involving these functions or differential equations where trigonometric identities are utilized. By recognizing these connections, students can enhance their problem-solving toolkit for tackling complex calculus challenges.

Related terms

L'Hôpital's Rule:

A method for finding limits of indeterminate forms by differentiating the numerator and denominator.

Taylor Series: A representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point.

Sine and Cosine Limits: Limits involving sine and cosine functions that often result in key values, such as 0 or 1, as x approaches 0.